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MACHINE DESIGN 



BY 
CHARLES H.' BENJAMIN 

DEAN OF THE SCHOOLS OF ENGINEERING, PURDUE UNIVERSITY 
\ 

AND 

JAMES D. HOFFMAN 

PROFESSOR OF MECHANICAL ENGINEERING AND PRACTICAL 
MECHANICS, UNIVERSITY OF NEBRASKA 




NEW YORK 
HENRY HOLT AND COMPANY 

1913 






<^A 



.< 



Copyright, 1909 

By 

JAMES D. HOFFMAN 



Copyright, 1906, 1913 

By 

HENRY HOLT AND COMPANY 



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THE. MAPLE. PRESS. YORK. PA 



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©CI,A332597 



PREFACE 

The present book represents the consolidation of two texts 
on this subject, Benjamin's Machine Design and Hoffman's 
Elementary Machine Design. 

As now arranged, the book serves two purposes: That of a 
text for the classroom, embodying the theory and practice of 
design, and that of a reference book for the drafting room, 
illustrating the design of complete machines. 

The authors recognize the fact that there are two methods of 
teaching this subject, one by details separately treated as ele- 
ments, one by a consideration of the complete machine, i.e., one 
method is synthetic and one analytic. It is believed that this 
book will afford a means of using either method or both combined. 

Some important additions to the text are worthy of mention. 
Chapter II, on Materials, has been rewritten. Much additional 
matter on the subject of cast-iron frames has been introduced, 
involving the results of numerous experiments. The theoretical 
and experimental strength of steel tubes under collapsing pres- 
sures is quite fully discussed and additional data are given on 
the failure of pipe fittings. 

Other subjects which receive in this volume fuller treatment 
than heretofore are Flat plates, Crane hooks. Leaf springs. Bear- 
ings, both plain and rolling, Clutches, Gear teeth and Belting. 



in 



TABLE OF CONTENTS 

Chapter Page 

Inteoduction. — Units and Formulas 1 

1. Units. 2. Abbreviations. 3. Notation. 4. Formulas. 5. 
Profiles of uniform strength. 

I. Materials 9 

6. Primary classification. 7. Iron. 8. Steel. 9. Steel alloys. 
10. Copper alloys. 11. Strength and Elasticity. 

II. Frame Design 21 

12. General principles of design. 13. Machine supports. 14. 
Machine frames. 15. Tests on simple beams. 16. Shapes of 
frames. 17. Stresses in frames. 18. Professor Jenkin's ex- 
periments. 19. Purdue tests. 20. Principles of design. 

III. Cylinders and Pipes 50 

21. Thin shells. 22. Thick shells. 23. Steel and wrought iron 
pipe. 24. Strength of boiler tubes. 25. Theory. 26. Tube 
joints. 27. Tubes under concentrated loads. 28. Pipe fit- 
tings. 29. Flanged pittings. 30. Steam cylinders. 31. Thick- 
ness of flat plates. 32. Steel plates. 

IV. Fastenings 91 

• 33. Bolts and nuts. 34. Crane hooks. 35. Riveted joints. 
36. Lap joints. 37. Butt joint with two straps. 38. Efl&- 
ciency of joints. 39. Butt joints with unequal straps. 40. 
Practical rules. 41. Riveted joints for narrow plates. 42. 
Joint pins. 43. Cotters. 

V. Springs 107 

44. Helical springs. 45. Square wire. 46. Experiments. 47. 
Springs in torsion. 48. Flat springs. 49. ElKptic and semi- 
elKptic springs. 

VI. Sliding Bearings 120 

50. Slides in general. 51. Angular slides. 52. Gibbed sHdes. 
53. Flat sHdes. 54. Circular guides. 55. Stuffing boxes. 

VII. Journals, Pivots and Bearings 128 

56. Journals. 57. Adjustment, 58. Lubrication. 59. Fric- 
tion of journals. 60. Limits of pressure. 61. Heating of jour- 
nals. 62. Experiments. 63. Strength and stiffness of jour- 
nals. 64. Caps and bolts. 65. Step bearings. 66. Friction of 
pivots. 67. Flat collar. 68. Conical pivot. 69. Schiele's 
pivot. 70. Multiple bearing. 

V 



vi TABLE OF CONTENTS 

Chaptkb Page 

VIII. Ball and Roller Bearings 153 

71. General principles. 72. Journal bearings. 73. Step bear- 
ings. 74. Materials and wear. 75. Design of bearings. 76. 
Endurance of ball bearings. 77. Roller bearings. 78. Grant 
roller bearings. 79. Hyatt rollers. 80. Roller step bearings. 

81, Design of roller bearings. 

IX. Shafting, Couplings and Hangers 167 

82. Strength of shafting. 83. Combined tension and bending. 
84. Couplings. 85. Clutches. 86. Automobile clutches. 87. 
Coupling bolts. 88. Shafting keys. 89. Strength of keyed 
shafts. 90. Hangers and boxes. 

X. Gears, Pulleys and Cranks 186 

91. Gear teeth. 92. Strength of gear teeth. 93, Lewis' for- 
mula. 94, Experimental data. 95. Modern practice. 96. 
Teeth of bevel gears. 97. Rim and arms. 98. Sprocket wheels 
and chains. 99. Silent chains, 100, Cranks and levers. 

XI. Fly-wheels 204 

101. In general. 102, Safe speed for wheels. 103. Experi- 
ments on fly-wheels. 104. Wooden pulleys, 105, Rims of cast- 
iron gears. 106, Rotating discs, 107. Plain discs. 108. Con- 
ical discs. 109. Discs with logarithmic profile, 110, Bursting 
speeds. 111. Tests of discs. 

XII. Transmission by Belts and Ropes 221 

112. Friction of belting. 113. SHp of belt. 114. Coeflicient 
of friction. 115. Strength of belting, 116, Taylor's experi- 
ments, 117, Rules for width of belts. 118, Speed of belting, 
119. Manila rope transmission. 120, Strength of Manila ropes. 
121. Cotton rope transmission. 122. Wire rope transmission. 

XIII. Design of Toggle-joint Press 235 

123. Introductory. 124. Drawings. 125. Calculations, 126. 
Analysis of forces. 127. Design of lever, 128. Shapes of 
levers, 129. Hole in lever, 130. Fastening for standard. 
131. Design of standard. 132. Toggle joint. 133. Shapes of 
toggle members. 134. Die heads. 135. Frame or bed. 136. 
Stabihty of frame, 137. Toggle press. 138. Vertical hand- 
power press. 139. Foot-power press. 140. Hand-power punch. 

141. Punch and shear. 

XIV. Design of Belt-driven Punch or Shear 265 

142. General statement. 143. Requirements of design. 144. 
Design of frame. 145. Outline of frame. 146. Shearing force. 
147. Depth of cut. 148. Sizes of pulleys. 149. Weight of 
fly-wheel. 150. Driving shaft. 151. Gears. 152. Main shaft. 
153. SHding head. 154. Clutches and transmission devices. 



TABLE OF CONTENTS vii 

Chapter Page 

155. Punch, die and holders. 156. The bevel shear. 157. 
Horizontal power punch. 158. The bull-dozer. 159. Power 
press. 160. Rotary shear. 161. Sheet metal Sanger. 162. 
Flanging machine. 

XV. Design of Air Hoist and Riveter 299 

163. Air hoist. 164, 165. Portable riveter. 166. Alligator 
riveter. 167. Mud-ring riveter. 168. Lever riveter. 189. 
Hydraulic riveter. 170. Triple pressure hydrauhc riveter. 

XVI. Studies in the Kinematics of Machines 309 

171. Gear tooth outUnes. 172. Planer cam. 173. Sewing 
machine cam. 174. Sewing machine bobbin winder. 175. 
Constant diameter cam. 176, 177, 178, 179. Quick return 
motions. 180. Cam and oscillating arm. 181. Two-motion 
cam. 182. Conical cam. 183, 184. Motion problems. 185. 
Forming machine for wire clips. 186-202. Motion problems. 
203. Mechanism of inertia governor. 204. Mechanism of 
centrifugal governors. 205. Straight line governor. 206. 
Stephenson Hnk motion. 207. Walschaert valve gear. 
Index , 336 



TABLES 

Table Page 

I. Values of q in Column Formula 4 

I a. Values of S and k in Column Formula 5 

II. Constants of Cross-section 6 

III. Formulas for Loaded Beams 7 

IV. Classification of Metals 9 

V. Composition of Bronzes 16 

VI. Strength of Wrought Metais 18 

VII. Strength of Cast Metals 19 

VIII. Strength of Cast Iron Beams 30 

IX. Strength of Cast Iron Beams 32 

X. Strength of Cast Iron Beams . 33 

XI. Strength of Cast Iron Beams 35 

XII. Stresses in Machine Frames 38 

XIII. Strength of Riveter Frames 41 

XIV. Strength of Riveter Frames 44 

XV. Sizes of Iron and Steel Pipe 56 

XVI. Sizes of Extra Strong Pipe 58 

XVII. Sizes of Double Extra Strong Pipe 59 

XVIII. Sizes of Iron and Steel Boiler Tubes 60 

XIX. Collapsing Pressure of Tubes 64 

XX. Stiffness of Steel Hoops 70 

XXI. Strength of Standard Screwed Pipe Fittings ... 72 

XXII. Strength of Flanged Fittings 74 

XXIII. Bursting Strength of Cast Iron Cylinders .... 80 

XXIV, Strength of Reinforced Cylinders 82 

XXV. Strength of Cast Iron Plates 86 

XXVI. Strength of Cast Iron Plates 87 

XXVII. Stresses in Flat Plates 89 

XXVIII. Strength of Iron or Steel Bolts 91 

XXIX. Dimensions of Machine Screws 94 

XXX. Elastic Limit of Crane Hooks 94 

XXXI. Dimensions of Riveted Lap Joints 101 

XXXII. Dimensions of Riveted Butt Joints 101 

XXXIII. Strength and Stiffness of Helical Springs .... 110 

XXXIV. Friction of Piston Rod Packings 126 

XXXV. Friction of Piston Rod Packings 126 

XXXVI. Friction of Piston Rod Packings 127 

XXXVII. Friction of Journal Bearings 139 

XXXVIII. Tests of Large Journals 140 

ix 



PLATES 

Table Page 

XXXIX. Marine Thrust Bearings 151 

XL. Friction of Roller and Plain Bearings 161 

XLI. Coefficients op Friction 161 

XLII. Coefficients of Friction 162 

XLIII. Safe Loads for Roller Bearings 164 

XLIV. Safe Loads for Roller Step Bearings 165 

XLV. Values of k for Roller Thrust Bearings .... 166 

XL VI. Diameters of Shafting 168 

XL VII. Power of Clutches 177 

XL VIII. Efficiency of Keyed Shafts 181 

XLIX. Proportions of Gear Teeth 187 

L. Sizes of Test Fly-wheels 208 

LI. Sizes of Test Fly-wheels 209 

LII. Flanges and Bolts of Test Fly-wheels 209 

LIII. Failure of Flanged Joints 210 

LIV. Sizes of Linked Joints 210 

LV. Failure of Linked Joints 210 

LVI. Bursting Speeds of Rotating Discs 218 

LVII. Bursting Speeds of Rotating Discs 219 

LVIII. Horse-power of Manila Rope 231 

LIX. Horse-power of Cotton Rope 232 

LX. Horse-power of Wire Rope 233 



plates 

Plates Page 

C 1. Toggle Joint Press Assembly Facing 238*^ 

C 2. Toggle Joint Press Details Facing 238 '' 

C 3. Toggle Joint Press Details Facing 240*^ 

C 4. Single Power Punch Assembly 287'^ 

C 5. Single Power Punch Details 288"^ 

C 6. Single Power Punch Details 289^ 

C 7. Single Power Punch Details Facing 290 ^^ 

C 8. Single Power Punch Details 290 */ 



MACHINE DESIGN 



MACHINE DESIGN 



INTRODUCTION 

UNITS AND FORMULAS 

1. Units. — In this book the following units will be used unless 
otherwise stated. 

Dimensions in inches. 

Forces in pounds. 

Stresses in pounds per square inch. 

Velocities in feet per second. 

Work and energy in foot pounds. 

Moments in pounds inches. 

Speeds of lotation in revolutions per minute. 

The word stress will be used to denote the resistance of material 
to distortion per unit of sectional area. The word deformation 
will be used to denote the distortion of a piece per unit of length. 
The word set will be used to denote total permanent distortion. 

In making calculations the use of the slide-rule and of four- 
place logarithms is recommended; accuracy is expected only to 
three significant figures. 

2. Abbreviations. — The following abbreviations are among 
those recommended by a committee of the American Society of 
Mechanical Engineers in December, 1904, and will be used 
throughout the book.^ 

NAME ABBREVIATION 

Inches in. 

Feet . ft. 

Yards. yd. 

' Tr. A. S. M. E., Vol. XXVI, p. 60. 

1 



2 MACHINE DESIGN 

NAME ABBREVIATION 

Miles spell out. 

Pounds lb. 

Tons spell out. 

Gallons gal. 

Seconds sec. 

Minutes min. 

Hours hr. 

Linear lin. 

Square sq. 

Cubic cu. 

Per spell out. 

Fahrenheit f ahr. 

Percentage % or per cent. 

Revolutions per minute r.p.m. 

Brake horse power b.h.p. 

Electric horse power e.h.p. 

Indicated horse power i.h.p. 

British thermal units B.t.u. 

Diameter Diam. 



3. Notation. 




Arc of contact 


= radians, 


Area of section 


= A sq. in. 


Breadth of section 


= h in. 


Coefficient of friction 


=/ 


Deflection of beam 


= A in. 


Degrees 


= deg. 


Depth of section 


= hm. 


Diameter of circular section 


= d in. 


Distance of neutral axis from outer fiber 


= y in. 


Elasticity, modulus of, 




in tension and compression 


= E 


in shearing and torsion 


= G 


Heaviness, weight per cu. ft. 


= w 


Length of any member 


= 1 in. 


Load or dead weight 


= Wlh. 


Moment, in bending 


= Mlb.-in. 


in twisting 


= T Ib.-in. 



FORMULAS 


3 


Moment of inertia 




rectangular 


= 1 


polar 


=J 


Pitch of teeth, rivets, etc. 


= p in. 


Radius of gyration 


= r in. 


Section modulus, bending 


y 


twisting 


y 

=s 


Stress per unit of area 


Velocity in feet per second 


— V ft. per sec. 


4. Formulas. 




Simple Stress 





w 

Tension, compression or shear, ^=—7 (1) 

Bending under Transverse Load 

SI 
General equation, M= — (2) 

Rectangular section, ilf = — ^ — (3) 

Rectangular section, hh^ = ~^' (4) 

Sd^ 
Circular section, ilf = -— — . (5) 

Circular section, d = \ — '-^ (6) 

Torsion or Twisting 

O J 

General equation, T = — (7) 

Sd^ 
Circular section, T = -i^^' (8) 

5.1 

3/5.1 T 
Circular section, d = \— -o — (9) 

Hollow circular section, T = ^~ — -1— ^- (10) 

o. i. a 

Other values of - and — may be taken from Table II. 

y y 



4 MACHINE DESIGN 

Combined Bending and Twisting 

Calculate shaft for a bending moment, 

T' = i{M + \/M' + T'). 

Column subject to Bending 
W S 



Use Rankine's formula, 



A 



1+3 



(11) 



(12) 



The values of r^ may be taken from Table II. The subjoined 
table gives the average values of q, while S is the compressive 
strength of the material. 



TABLE I 

Values of q in Formula 12 



Material 


Both ends 
fixed 


Fixed and 
round 


Both ends 
round 


Fixed and 
free 


Timber 


1 

3000 

1 

5000 

1 


1.78 
3000 

1.78 
5000 

1.78 
36000 

1.78 
25000 


4 
3000 

4 
5000 

4 


16 
3000 

16 

5000 

16 
36000 

16 
25000 


Cast iron 

Wrought iron .... 
Steel 


36000 
1 


36000 
4 


25000 


25000 



Carnegie's hand-book gives iS = 50,000 for medium steel 

for the three first 



columns and g=36wo, 24000 and tswo- 
columns in above table. 

In this formula, as in all such, the values of the constant 
should be determined for the material used by direct experiment 
if possible. 

W I 

Or use straight Ime formula, -j-=S—k-' (12a) 



COLUMNS 



TABLE la 

Values of S and k in Formula (12a) 
(Merriman's Mechanics of Materials) 



Kind of column 


S 


k 


Limit - 
r 


Wrought Iron: 

Flat ends 


42,000 
42,000 
42,000 

52,500 
52,500 
52,500 

80,000 
80,000 
80,000 

5,400 


128 
157 
203 

179 
220 

284 

438 
537 
693 

28 


218 
178 
138 

195 
159 
123 

122 
99 

77 

128 


Hinged ends 


Round ends 


Mild Steel: 

Flat ends 


Hinged ends 


Round ends 


Cast Iron: 

Flat ends 


Hinged ends 


Round ends 


Oak: 

Flat ends 





Carnegie's hand-book gives allowable stress for structural 
columns of mild steel as 12,000 for lengths less than 90 radii, and 

as 17,100 —57 - for longer columns. 

This allows a factor of safety of about four. 



6 



MACHINE DESIGN 



TABLE II 
Constants of Cross-section 



Form of 

section and 

area A 


Square of 
radius of 
gyration 

r2 


Moment 

of 

inertia 

I = Ar^ 


Section 

modulus 

/ 

y 


Polar 

moment 

of inertia 

J 


Torsion 

modulus 

J 

y 


Rectangle 
bh 

Square 
d^ 

HoUow 

rectangle or 

/—beam 

bh — bihi 

Circle 
Id^. 

Hollow 
circle 

^(d^-dh) 
Ellipse 


12 

d^ 
12 

bh^-bihh 
12{bh-bihi) 

d2 
16 

d^+dh 
16 

a2 
16 


bh^ 
12 

12 

bh^-^bihh 
12 

7rd4 
64 

nidi-dh) 
64 

i:ba^ 
64 


bh^ 
6 

6 

bh^-bWi 
6h 

d3 
10.2 

di-dh 
10.2d 

ba^ 
10.2 


bh3+ bm 
12 

di 
6 

7:di 
32 

7:(d*-dh) 
32 

;r(6o3+ o63) 
64 


bh^+b^h 


6^62+^2 

4.24 
5.1 

d4-(Z4i 

5.1d 

6a3+a63 
10.2a 



Values of I and J for more complicated sections can be worked out from 
those in table. 



LOADED BEAMS 



TABLE III 

Formulas for Loaded Beams 



Beams of uniform cross-section 


Maximum 

moment 

M 


Maximum 
deflection 

A- 


Cantilever, load at end 


Wl 

Wl 

2 

Wl 
4 

Wl 

8 

3TFZ 
16 

Wl 

8 

Wl 

8 

Wl 
12 

Wl 
2 


WP 
SEI 

WP 
SEI 

WP 
48^/ 

5WP 

dS4:EI 

.0182WZ* 
EI 

.00541^^3 

EI 

WP 
192EI 

WP 
S84.EI 

WP 
12EI 


Cantilever, uniform load 


Simple beam, load at middle 


Simple beam, uniform load 


Beam fixed at one end, supported at other, 
load at middle. 

Beam fixed at one end, suppported at other, 
uniform load. 

Beam fixed at both ends, load at middle . . . 

Beam fixed at both ends, uniform load 

Beam fixed at both ends, load at one end, 
(pulley arm). 



5. Profiles of Uniform Strength. — In a bracket or beam of 
uniform cross-section the stress on the outer row of fibers in- 
creases as the bending moment increases and the piece is most 
liable to break at the point where the moment is a maximum. 
This difficulty can be remedied by varying the cross-section in 
such a way as to keep the fiber stress constant along the top or 
bottom of the piece. The following table shows the shapes to 
be used under different conditions. 



8 



MACHINE DESIGN 



Type 


Load 


Plan 


Elevation 


Cantilever 

Cantilever 

Simp. Beam 

Simp, beam 


Center .... 
Uniform. . 
Center .... 

Uniform. . 


Rectangle . . . 
Rectangle . . . 
Rectangle . . . 

Rectangle . . . 


Parabola, axis horizontal. 

Triangle. 

Two parabolas intersecting 
under load. 

Ellipse, major axis hori- 
zontal. 



The material is best economized by maintaining a constant 
breadth and varying the depth as indicated. 

This method of design is applicable to cast pieces rather than 
to those that are forged or cut." 

The maximum deflection of cantilevers and beams having a 
profile of uniform strength is greater than when the cross-section 
is uniform, 50 per cent greater if the breadth varies, and 100 
per cent greater if the depth varies. 



CHAPTER 1 



MATERIALS 



6. Primary Classification. — The materials used in machine 
construction are practically all metals. They may be classified 
in two ways : (a) According to the principal metallic constituents 
such as iron, copper, tin, etc. ; (b) as cast or wrought metals 
according to the methods employed in preparing them for use. 

The following table combines the two methods of classification. 

TABLE IV 



Principal metal 


Cast 


Wrought 


Iron 

Copper < 

Tin 

Aluminum . 


Cast iron 

Malleable iron 

Steel castings 

Bronze 

Brass 

Babbitt metal 

Bronze 


Wrought iron. 
Soft steel. 
Tool steel. 
Alloy steel. 
Brass wire. 
Sheet brass. 

Rolled or drawn. 



7. Iron. — Commercial iron is produced from iron ore by 
reduction in a blast furnace. Most iron ores are oxides and also 
contain earthy impurities such as silica and alumina. 

The oxygen is removed by the burning of the coke used as 
fuel, while the limestone used as a flux unites with the silica 
and alumina forming a glassy slag which floats on the molten 
iron. 

Pig Iron. — The coarse-grained impure iron thus formed is 
the pig iron of commerce and from it is made ordinary cast iron 
by remelting in the cupola of the foundry. Pig iron contains 
besides iron various quantities of carbon, silicon, manganese, 
phosphorus and sulphur. The last two are impurities and if 

9 



10 MACHINE DESIGN 

present in any considerable quantity render the pig unsuitable 
for the manufacture of high-grade irons or steels. The phos- 
phorus comes from the ore and the sulphur from the fuel used. 
The use of high-grade ore and of coke made from a non-sulphur 
coal is necessary to the production of pure iron. Pig iron may 
be used in the foundry for the manufacture of iron castings, in 
the puddling mill for producing wrought iron, or in the steel 
mill for the manufacture of Bessemer or of open-hearth steel. 

Cast Iron. — Iron castings are made in the foundry by melting 
pig iron in a cupola using coke for a fuel. The quality of the 
cast iron depends largely upon the character of the pig iron used, 
as there is little chemical change affected in the cupola. A 
certain amount of scrap cast iron may be melted with the 
charge; remelting of iron makes it finer grained and harder. 
Wrought iron or steel shavings mixed with the molten cast iron 
produces a tough fine-grained iron, sometimes called semi-steel. 
The addition of about 25 per cent of steel scrap makes a fine- 
grained soft iron having a tensile strength about 50 per cent 
greater than that of the cast iron without the steel. 

Carbon exists in cast iron in two forms: (a) chemically com- 
bined with the iron; (h) as free carbon or graphite. The larger 
the proportion of free carbon, the softer and weaker is the iron. 
Remelting and cooling increases the amount of combined carbon 
and makes the iron harder as before noticed. The total amount 
of carbon present varies from 2 to 5 per cent in different irons. 

Silicon is an important element in iron and influences the rate 
of cooling. The more slowly iron cools after melting, the more 
graphite forras, the less the shrinkage and the softer the iron. 
Two per cent of silicon gives a soft gray iron with a high tensile 
strength. Machinery iron contains usually from IJ to 2 per 
cent of silicon. 

Chilled iron is cast iron which has been cooled suddenly in the 
mold by contact with metal or some other good conductor of 
heat. Chilling increases the amount of combined carbon and 
makes the iron white and hard. It is used on surfaces which 
need to be extremely hard and durable, as the treads of car 
wheels and the outside of the rolls used on steel mills. The 
depth of the chill depends on the amount of metal used in the 
cooling surface of the mold. 



CAST IRON 11 

All castings are chilled slightly on the surface. An examina- 
tion of a freshly fractured casting shows whiter and finer-grained 
metal around the edges than at the center. For this reason, 
castings having considerable surface or " skin " in proportion to 
their weight are relatively stronger (see Art. 15). 

In selecting cast iion for various machine members, soft gray 
irons should be chosen where workability rather than strength is 
desired. Medium gray irons having a fine grain should be used 
where moderate strength and hardness are necessary as in the 
cylinders of steam engines and pumps. Hard gray iron is only 
suitable for heavy castings which require little or no machining, 
as it is brittle and not easily worked. An examination of the 
fracture of a sample of iron is a guide in determining its desira- 
bility for any particular case. 

Cast iron is the cheapest and best material for pieces of irregu- 
lar and complicated shape; it has a high compressive and a low 
tensile strength; it is brittle and cannot be welded or forged; 
but it resists corrosion much better than wrought iron. For its 
use in machine construction, see Art. 14. 

Malleable Iron, — Malleable iron is cast iron annealed and 
partially decarbonized by being heated in an annealing oven 
in contact with some oxidizing material such as hematite ore, 
and then being allowed to cool slowly. A white cast iron is 
best for this process as the presence of graphitic carbon interferes 
with its success. An iron containing a small amount of silicon 
and considerable manganese promotes the formation of combined 
carbon just as silicon promotes the formation of free carbon. 

The castings before being annealed are hard and brittle, the 
fracture showing a silvery appearance. They are packed in 
air-tight cast-iron boxes with the oxidizing material and are 
kept at a red heat for several days. They are allowed to cool 
slowly and when removed are tough and ductile with a dull 
gray fracture. 

The oxidation removes some of the total carbon from the 
surface of the material and the heating and slow cooling changes 
the most of that remaining to graphite. 

An iron which originally contains 2.8 per cent combined and 
0.20 per cent free carbon, after annealing may show 0.20 per 
cent combined and 1.8 per cent free carbon. 



12 MACHINE DESIGN . 

Malleable castings are particularly suitable for small parts 
having irregular shapes. The metal does not possess as much 
ductility or tensile strength as wrought iron but occupies a place 
intermediate between that and cast iron. 

As the process of malleablizing is to a certain extent a super- 
ficial one, it is best adapted to thin metal, although castings an 
inch or more in thickness have been successfully treated. 

Wrought Iron. — Wrought iron is commercially pure iron which 
is made from pig iron by decarbonizing it in the puddling furnace. 
This furnace is a reverberatory one in which the molten pig is 
subjected to the action of the hot gases from the fuel. 

The silicon, manganese and carbon are oxidized or burned 
out, either by the action of the gas or by oxide of iron introduced 
with the charge. A part of the phosphorus and sulphur is also 
oxidized in the puddling. The molten mass is continually 
stirred during the process and finally assumes a pasty consis- 
tency. It is then squeezed to remove the slag and rolled 
into bars. These are cut, piled and welded into either bar or 
plate iron. 

The particles of iron in the puddling process are more or less 
enveloped in the slag and as the mass is squeezed and rolled, 
these particles become fibers separated from each other by a thin 
sheath or covering of slag, and it is this which gives wrought 
iron its characteristic structure. 

The presence of either sulphur or phosphorus in the iron 
renders it less reliable. 

Wrought iron possesses moderate tensile strength and high 
ductility. It can be forged and welded readily. Hammering 
or rolling it cold increases its strength and stiffness to a certain 
degree and raises artificially its elastic limit. For most purposes, 
it has been replaced of late years by soft steel. Either of these 
metals may be rendered superficially hard by the process known 
as case hardening. The pieces to be treated are packed in air- 
tight boxes together with pulverized carbon in some form, 
usually bone-black. The boxes are brought to a red heat and 
kept so for several hours. The pieces are then removed and 
quenched suddenly in water. The surface of the iron has com- 
bined with the carbon in which it was packed and changed to a 
high-carbon or hardening steel. Such pieces have a soft, ductile 



STEEL 13 

center and a hard surface. Case hardening can be done after 
finishing but is liable to warp the metal. 

8. Steel. — Steel is made from molten pig iron by burning out 
the silicon and carbon with a hot blast, either passing through 
the liquid as in the Bessemer converter, or over its surface as in 
the open-hearth furnace. A suitable quantity of carbon and 
manganese is then added and the metal poured into ingot molds. 
If the ingots are reheated and rolled, structural steel and rods 
or rails are the result. 

Manganese has the effect of preventing blowholes and giving 
the steel a more uniform texture. 

Open-hearth steel differs but little from Bessemer in its chemical 
composition but is more uniform in quality on account of the 
more deliberate nature of the process of manufacture. Boiler 
plate, structural steel, and in general, material which is respon- 
sible for the safety of life and limb should be of open-hearth 
rather than Bessemer steel. 

Steel containing not more than 0.6 per cent of carbon is known 
as soft steel. It has a higher elastic limit and greater tensile 
strength than wrought iron, which metal it has practically sup- 
planted in the manufacture of machine parts. It is very ductile 
and malleable and may be welded if not too high in carbon. 

Crucible steel is made by melting steel or a mixture of iron and 
carbon in a crucible and pouring the melted metal into molds, and 
hence is commonly known as cast steel. 

This method is used for producing the harder steels suitable 
for cutting tools. The amount of carbon will vary from 0.5 to 
1.5 per cent according to the use to be made of the steel. Such 
steel contains small amounts of silicon and manganese but must 
be practically free from sulphur and phosphorus. 

It is relatively high priced and is not used for ordinary machine 
parts. It cannot be readily welded but possesses the very useful 
characteristic of hardening when heated to a red heat and cooled 
suddenly. The degree of hardness can be controlled by accu- 
rately measuring the temperature of heating and by using various 
cooling agents such as water, brine and different kinds of oil. 
The steel can be tempered or softened after hardening by reheat- 
ing to a slight degree. 



14 MACHINE DESIGN 

In machine construction crucible steel is only used for screws, 
spindles, ratchets, etc., which need to be extremely hard. It 
has a high tensile and compressive strength but is brittle and 
liable to contain hardening cracks. 

Steel castings are made by pouring fluid open-hearth steel 
directly into molds. They possess somewhat the same charac- 
teristics as malleable castings, being relatively tough and 
ductile. 

It has been somewhat difficult in the past to obtain reliable 
castings of this material as the great shrinkage' — about double 
that of cast iron — has tended to make them porous and spongy 
in spots. 

Furthermore, steel which was sufficiently low in carbon to 
make soft castings was not fluid enough to run sharply in the 
mold. 

These difficulties have been to a large extent overcome and 
it is now possible to obtain steel castings which are reasonably 
clean and sound. They have about the same chemical composi- 
tion as mild rolled steel, the carbon varying from 0.2 to 0.6 per 
cent, the silicon about the same and manganese from 0.5 to 1 
per cent. Steel castings when first poured are coarse-grained 
and should be annealed to make them tough and ductile. 

9. Steel Alloys. — Steel alloys are compounds of steel with 
chromium, vanadium, manganese, etc.; strictly speaking, all 
steels are alloys of iron with other substances, but when the term 
steel is used without qualification, it is understood to mean 
carbon steel. 

Nickel steel is both stronger and tougher than carbon steel. 
A high carbon steel is strong but brittle; the same or greater 
strength can be obtained by the addition of nickel without 
materially diminishing the ductility. This metal is suitable for 
pieces which are subject to severe shocks. 

Manganese steel is an alloy containing about 1 per cent of 
carbon and from 10 to 20 per cent of manganese; 14 per cent of 
manganese gives the maximum of strength and ductility com- 
bined. This metal is strong, tough and extremely hard, so that 
it cannot be readily finished except by grinding. It can be 
used for cutting tools, and like nickel steel is valuable for pieces 



ALLOY STEELS 15 

subjected to great stress and wear. Its strength is increased by 
heating and sudden cooling. 

Chromium is sometimes added to nickel steel in the manu- 
facture of safes and armor plate. 

Mushet steel is an alloy of high carbon steel with tungsten and 
manganese and was the first of the air-hardening steels used for 
cutting tools. Like all of this class of tool steels, it must be 
worked at a yellow heat and hardens when cooled slowly in the 
air. 

The so-called air-hardening or high-speed steels are of various 
chemical compositions, containing carbon, manganese, tungsten, 
chromium, molybdenum or titanium, but the exact ingredients 
and proportions are for the most part trade secrets. Such 
steels are usually purchased in small sections and are used in 
special tool holders. They are forged with great difficulty and 
are generally heated in special furnaces with pyrometers for 
determining the exact temperature, and cooled in an air blast 
or by dipping in oil baths. The difference of a few degrees in the 
temperature of the metal will make or mar the cutting efficiency. 
They are of no use in machine construction, but affect it indirectly 
by requiring much greater strength, rigidity and power in 
machine tools. 

It is not an uncommon thing for the power consumption of a 
lathe or planer to be increased six or eight times by the use 
of the newer tools. 

Vanadium steel is one of the latest claimants for favor among 
the steel alloys. The addition of a small amount of this metal, 
0.1 or 0.2 per cent, increases the strength and stiffness of mild 
steel in a marked degree with comparatively little increase in 
its cost. 

It is already used extensively in machine construction, 
particularly in marine work. 

10. Copper Alloys. — These metals are alloys of copper and tin, 
copper and zinc or of all three. Copper is not used alone in 
machine construction except for electric conductors. Phos- 
phorus, aluminum and manganese are also used in combination 
with copper. 

The copper-tin alloys are commonly known as bronzes and are 



16 MACHINE DESIGN 

expensive on account of the large proportion of copper, from 
85 to 90 per cent. 

Copper-zinc alloys, on the other hand, are called brass, and for 
maximum strength and ductility should contain from 60 to 70 
per cent of copper. 

Bronzes high in tin and low in copper are weak, but have 
considerable ductility and make good metals for bearings. 
Tin 80, copper 10 and antimony 10 is Babbitt metal, so much 
used to line journal bearings, the antimony increasing the 
hardness. 

The late Dr. Thurston's experiments on the copper-tin-zinc 
alloys showed a maximum strength for copper 55, zinc 43 and 
tin 2 per cent. The tensile strength of this mixture was nearly 
70,000 lb. per square inch. 

Phosphor bronze is a copper alloy with a small amount of 
phosphorus added to prevent oxidation of the copper and thereby 
strengthen the alloy. 

Manganese bronze is an alloy of copper and manganese, usually 
containing iron and sometimes tin. A bronze containing about 
84 per cent copper, 14 per cent manganese and a little iron, 
has much the same physical characteristics as soft steel and 
resists corrosion better. 

There is practically no limit to the varieties of color, hardness, 
ductility and durability among the copper alloys. Some of the 
more common mixtures are here given. 

TABLE V 

Composition of Bronzes 



Name 


Composition 


Gun metal 

Bell metal 

Yellow brass 

Muntz metal 

Aluminum bronze 

Phosphor bronze 

Manganese bronze (1) . . . 

Manganese bronze (2) . . . 


Copper . 90, tin . 10 
Copper . 77, tin . 23 
Copper .65, zinc .35 
Copper .60, zinc .40 
Copper .90, aluminum .10 
Copper .89, tin .09, phosphorus .01 
Copper . 84, manganese . 14, iron . 02 
/Copper .675, manganese .18 
1 Zinc .13, aluminum .01, silicon .005 



FACTORS OF SAFETY 17 

11. Strength and Elasticity. — The constants for strength and 
elasticity given in the tables are only fair average values and 
should be determined for any special material by direct experi- 
ment when it is practicable. Many of the constants are not 
given in the table on account of the lack of reliable data for their 
determination. 

The strength of steel, either rolled or cast, depends so much 
upon the percentages of carbon, phosphorus and manganese, 
that any general figures are liable to be misleading. Structural 
steel usually has a tensile strength of about 65,000 lb. per square 
inch, while boiler plate usually has less carbon, a low tensile 
strength and good ductility. 

Factors of Safety. — A factor of safety is the ratio of the ultimate 
strength of any member to the ordinary working load which 
will come upon it. This factor is intended to allow for: (a) 
Overloading either intentional or accidental, (b) Sudden blows 
or shocks, (c) Gradual fatigue or deterioration of material. 
(d) Flaws or imperfections in the material. 

To a certain extent the term '' factor of ignorance " is justifiable 
since allowance is made for the unknown. Certain fixed laws 
may guide one, however, in making the selection of a factor. 
It is a well-known fact that loads in excess of the elastic limit are 
liable to cause failure in time. Therefore, when the elastic 
limit of the material can be determined, it should be used as a 
basis rather than to use the ultimate strength. 

Furthermore, suddenly applied loads will cause about double 
the stress due to dead loads. These considerations indicate four 
as the least factor that should be used when the ultimate strength 
is taken as a basis. Pieces subject to stress alternately in oppo- 
site directions should have large factors of safety. 

The following table shows the factors used in good practice 

under various conditions: 

Structural steel in buildings 4 

Structural steel in bridges 5 

Steel in machine construction 6 

Steel in engine construction 10 

Steel plate in boilers 5 

Cast iron in machines 6 to 15 

Castings of bronze or steel should have larger factors than 
rolled or forged metal on account of the possibility of flaws. 



18 



MACHINE DESIGN 



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20 MACHINE DESIGN 

Cast iron should not be used in pieces subject to tension or 
bending if there is a liability of shocks or blows coming on the 
piece. 

Note. — In giving references to transactions and periodicals, the follow- 
ing abbreviations will be used: 

Transactions of American Society of Mechanical \ ™ a a at tt 

Engineers, J 
American Machinist Am. Mach. 

Gassier' s Magazine Cass. 

Engineering Magazine Eng. Mag. 

Engineering News Eng. News 

Machinery Mchy. 

REFERENCES 

Materials of Machines. A. W. Smith. 
Mechanics of Materials. Merriman. 
Materials of Engineering. Thurston. 



CHAPTER II 

FRAME DESIGN 

12. General Principles of Design. — The working or moving 
parts should.be designed first and the frame adapted to them. 

The moving parts can be first arranged to give the motions 
and velocities desired, special attention being paid to compact- 
ness and to the convenience of the operator. 

Novel and complicated mechanisms should be avoided and 
the more simple and well-tried devices used. 

Any device which is new should be first tried in a working 
model before being introduced in the design. 

The dimensions of the working parts for strength and stiffness 
must next be determined and the design for the frame completed. 
This may involve some modification of the moving parts. 

In designing any part of the machine, the metal must be put 
in the line of stress and bending avoided as far as possible. 

Straight lines should be used for the outlines of pieces exposed 
to tension or compression, circular cross-sections for all parts in 
torsion, and profile of uniform fiber stress for pieces subjected 
to bending action. 

Superfluous metal must be avoided and this excludes all 
ornamentation as such. There should be a good practical 
reason for every pound of metal in the machine. 

An excess of metal is sometimes needed to give inertia and 
solidity and prevent vibration, as in the frames of machines 
having reciprocating parts, like engines, planers, slotting ma- 
chines, etc. 

Mr. Oberlin Smith has characterized this as the ''anvil" 
style of design in contradistinction to the ''fiddle" style. 

In one the designer relies on the mass of the metal, in the 
other on the distribution of the metal, to resist the applied forces. 

A comparison of the massive Tangye bed of some large high- 
speed engines with the comparatively slight girder frame used 
in most Corliss engines, will emphasize this difference. 

21 



22 MACHINE DESIGN 

It may be sometimes necessary to waste metal in order to 
save labor in finishing, and in general the aim should be to 
economize labor rather than stock. 

The designers should be familiar with all the shop processes 
as well as the principles of strength and stability. The usual 
tendency in design, especially of cast-iron work, is toward 
unnecessary weight. 

All corners should be rounded for the comfort and convenience 
of the operator, no cracks or sharp internal angles left where 
dirt and grease may accumulate, and in general special attention 
should be paid to so designing the machine that it may be safely 
and conveniently operated, that it may be easily kept clean, 
and that oil holes are readily accessible. The appearance of a 
machine in use is a key to its working condition. 

Polished metal should be avoided on account of its tendency 
to rust, and neither varnish nor bright colors tolerated. The 
paint should be of some neutral tint and have a dead finish so 
as not to show scratches or dirt. 

Beauty is an element of machine design, but it can only be 
attained by legitimate means which are appropriate to the 
material and the surroundings. 

Beauty is a natural result of correct mechanical construction 
but should never be made the object of design. 

Harmony of design may be secured by adopting one type of 
cross-section and adhering to it throughout, never combining 
cored or box sections with ribbed sections. In cast pieces the 
thickness of metal should be uniform to avoid cooling strains, 
and for the same reason sharp corners should be absent. The 
lines of crystallization in castings are normal to the cooled surface 
and where two fiat pieces come together at right angles, the 
interference of the two sets of crystals forms a plane of weakness 
at the corner. This is best obviated by joining the two planes 
with a bend or sweep. 

Rounding the external corner and filleting the internal one 
is usually sufficient. Where two parts come together in such a 
way as to cause an increase of thickness of the metal there are 
apt to be ^^blow holes" or '^hot spots" at the junction due to 
the uneven cooling. 

'^Strengthening" flanges when of improper proportions or in 



FRAMES 



23 



the wrong location are frequently a source of weakness rather 
than strength. A cast rib or flange on the tension side of a 
plate exposed to bending, will sometimes cause rupture by crack- 




FiG. 1. — Old Planing Machine. An Example of Elaborate 

Ornamentation. 



ing on the outer edge. When a crack is once started rupture 
follows almost immediately. When apertures are cut in a 



24 



MACHINE DESIGN 



frame either for core-prints or for lightness, the hole or aperture 
hsould be the symmetrical figure, and not the metal that sur- 
rounds it, to make the design pleasing to the eye. 

The design should be in harmony with the material used and 
not imitation. For example, to imitate structural work either 
of wood or iron in a cast-iron frame is silly and meaningless. 




Machine design has been a process of evolution. The earlier 
types of machines were built before the general introduction of 
cast-iron frames and had frames made of wood or stone, paneled^ 
carved and decorated as in cabinet or architectural designs. 



FRAMES 25 

When cast-iron frames and supports were first introduced 
they were made to imitate wood and stone construction, so that 
in the earlier forms we find panels, moldings, gothic traceries 
and elaborate decorations of vines, fruit and flowers, the whole 
covered with contrasting colors of paint and varnished as 
carefully as a piece of furniture for the drawing-room. Relics of 
this transition period in machine architecture may be seen in 
almost every shop. One man has gone down to posterity as 
actually advertising an upright drill designed in pure Tuscan. 

13. Machine Supports. — The fewer the number of supports the 
better. Heavy frames, as of large engines, lathes, planers, etc., 
are best made so as to rest directly on a masonry foundation. 
Short frames as those of shapers, screw machines and milling 
machines, should have one support of the cabinet form. The use 
of a cabinet at one end and legs at the other is offensive to the eye, 
being inharmonious. If two cabinets are used provision should 
be made for a cradle or pivot at one end to prevent twisting of 
the frame by an uneven foundation. The use of intermediate 
supports is always to be condemned, as it tends to make the 
frame conform to the inequalities of the floor or foundation on 
what has been aptly termed the ^^caterpillar principle." 

A distinction must be made between cabinets or supports which 
are broad at the base and intended to be fastened to the founda- 
tion, and legs similar to those of a table or chair. The latter are 
intended to simply rest on the floor, should be firmly fastened to 
the machine and should be larger at the upper end where the 
greatest bending moment will come. 

The use of legs instead of cabinets is an assumption that the 
frame is stiff enough to withstand all stresses that come upon it, 
unaided by the foundation, and if that is the case intermediate 
supports are unnecessary. 

Whether legs or cabinets are best adapted to a certain machine 
the designer must determine for himself. 

Where two supports or pairs of legs are necessary under a 
frame, it is best to have them set a certain distance from the 
ends, and make the overhanging part of the frame of a parabolic 
form, as this divides up the bending moment and allows less 
deflection at the center. Trussing a long cast-iron frame with 



26 MACHINE DESIGN 

iron or steel rods is objectionable on account of the difference in 
expansion of the two metals and the liability of the tension nuts 
being tampered with by workmen. 

The sprawling double curved leg which originated in the time 
of Louis XIV and which has served in turn for chairs, pianos, 
stoves and finally for engine lathes is wrong both from a practical 
and esthetic standpoint. It is incorrect in principle and is 
therefore ugly. 

EXERCISE 

1. Apply the foregoing principles in making a written criticism of some 
engine or machine frame and its supports. 

(a) Girder frame of engine. 

(b) Tangye bed of air compressor. 

(c) Bed, uprights and supports of iron planing machine. 

(d) Bed and supports of engine lathe. 

(e) Cabinet of shaping or milling machine, 

(f) Frame of upright drill, 

14. Machine Frames. — Cast iron is the material most used but 
steel castings are now becoming common in situations where 
the stresses are unusually great, as in the frames of presses, 
shears and rolls for shaping steel. 

Cored vs. Rib Sections. — Formerly the flanged or rib section was 
used almost exclusively, as but a few castings were made from 
each pattern and the cost of the latter was a considerable item. 
Of late years the use of hollow sections has become more common; 
the patterns are more durable and more easily molded than those 
having many projections and the frames when finished are more 
pleasing in appearance. 

The first cost of a pattern for hollow work, including the cost 
of the core-box, is sometimes considerably more but the pattern 
is less likely to change its shape and in these days of many 
castings from one pattern, this latter point is of more importance. 
Finally, it may be said that hollow sections are usually stronger 
for the same weight of metal than any that can be shaped from 
webs and flanges. 

Resistance to Bending. — Most machine frames are exposed to 
bending in one or two directions. If the section is to be ribbed 
it should be of the form shown in Fig. 3. The metal being of 



FRAMES 



27 



nearly uniform thickness and the flange which is in tension 
having an area three or four times that of the compression flange. 
In a steel casting these may be more nearly equal. The hollow 
section may be of the shape shown in Fig. 4, a hollow rectangle 
with the tension side re-enforced and slightly thicker than the 
other three sides. The re-enforcing flanges at A and B may often 
be utilized for the attaching of other members to the frame as in 
shapers or drill presses. The box section has one great advantage 
over the I section in that its moment of resistance to side bending 





Fig. 3. 



Fig. 4. 



or to twisting is usually much greater. The double I or the U 
section is common where it is necessary to have two parallel 
ways for sliding pieces as in lathes and planers. As is shown in 
Fig. 5 the two I's are usually connected at intervals by cross 
girts. 

Besides making the cross-section of the 
most economical form, it is often desirable 
to have such a longitudinal profile as shall 
give a uniform fiber stress from end to 
end. This necessitates a parabolic or 
elliptic outline of which the best instance 
is the housing or upright of a modern iron 
planer. 

Resistance to Twisting. — The hollow cir- 
cular section is the ideal form for all frames or machine mem- 
bers which are subjected to torsion. If subjected also to bend- 
ing the section may be made elliptical or, as is more common, 
thickened on two sides by making the core oval. See Fig. 6. 
As has already been pointed out the box sections are in general 
better adapted to resist twisting than the ribbed or I sections. 




Fig. 5. 



28 



MACHINE DESIGN 




Fig. 6. 



Frames of Machine Tools. — The beds of lathes are subjected 
to bending on account of their own weight and that of the saddle 
and on account of the downward pressure on the tool when work 
is being turned. They are usually subjected to torsion on ac- 
count of the uneven pressure of the supports. The box section 

is then the best; the double I commonly 
used is very weak against twisting. The 
same principle would apply in designing 
the beds of planers but the usual method 
of driving the table by means of a gear 
and rack prevents the use of the box sec- 
tion. The uprights of planers and the 
cross rail are subjected to severe bending 
moments and should have profiles of uni- 
form strength. The uprights are also sub- 
ject to side bending when the tool is taking a heavy side cut near 
the top. To provide for this the uprights may be of a box sec- 
tion or may be reinforced by outside ribs. 

The upright of a drill press or vertical shaper is exposed to a 
constant bending moment equal to the upward pressure on the 
cutter multiplied by the distance 
from center of cutter to center 
of upright. It should then be 
of constant cross-section from 
the bottom to the top of the 
straight part. The curved or 
goose-necked portion should 
then taper gradually. 

The frame of a shear press or 
punch is usually of the G shape 
in profile with the inner fibers in 

tension and the outer in compression. The cross-section should 
be as in Fig. 3 or Fig. 4, preferably the latter, and should be 
graduated to the magnitude of the bending moment at each 
point. (See Fig. 7.) 




Fig. 7. 



15. Tests on Simple Beams. — In 1902, a series of experiments 
was made on cast-iron beams of various sections at the Case 
School of Applied Science. * The work was done by Messrs. 



CAST IRON BEAMS 29 

A. F. Kwis and R. H. West^ under the direction of the author 
and the results were reported by him in 1906. 

The patterns were all 20 in. long and had the same cross-section 
of 4.15 sq. in. As may be seen from the tables, the areas of the 
cast beams varied slightly. The castings of each set were all 
made from the same ladle of iron and were cast on end. A soft 
gray iron was used and a large flush basin distributed the molten 
metal to the mold, giving a uniform temperature and quality. 
The castings were prepared by Mr. Thomas D. West and proved 
to be remarkably uniform in quality and free from imperfections. 

The specimens were all tested by loading transversely at the 
center, the supports being 18 in. apart. 

Object. — The investigation had two distinct objects in view 
and two classes of test pieces were used. The first class com- 
prised Nos. 1 to 11 and Nos. 22 to 32, and these specimens had 
sections such as are used in parts of machines. 

The second class comprised Nos. 12 to 21 and 33 to 42, all 
having sections similar to those used in the rims of fly-wheels. 
The sections tested were such as shown by the diagrams in the 
tables. 

The areas given in the table are those of the specimens at the 
point of rupture. There are two specimens of each shape cast 
from the same pattern. 

The section modulus - was calculated from the dimensions 

y , , 
of the casting at the breaking point, y being the distance from 

neutral axis to extreme fiber in tension. In testing each specimen 
the load was applied gradually and readings of the deflection 
were taken at regular intervals. When the ^'set" load was 
reached, the pressure was removed and a reading of the perma- 
nent set was taken. The load was again applied and observations 
made on the deflection up to near the time of rupture. 

The load-deflection curves plotted from these observations 
are nearly all smooth and uniform in character, as may be seen 
by reference to Fig. 8 which shows the curves for No. 33. 

The initial line curves gradually from the start showing an 
imperfect elasticity, while the set line is nearly straight and 
approximately parallel to the tangent of the curve at the vertex. 

1 Mchy., May, 1906. 



30 



MACHINE DESIGN 



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CAST IRON BEAMS 31 

The so-called moduli of elasticity were calculated from the 
set lines using the formula 



48 A/ 



In each test a reading of the load was taken at the instant 
when the deflection measured 0.03 in., and these loads may be 
taken as a fair measure of the ^'stiffness" of the section. 

The modulus of rupture was calculated from the breaking load 
and the section modulus, using the formula: 

My Wly 

The modulus of rupture, as S is generally called, is supposed to 
represent the tensile stress on the outer fibers at the point of 
rupture and to measure in a way the transverse strength of the 
material. In the absence of a better measure we will use this, 
and take the circular and square sections as our standards. The 
average value of S for the four is 24,360 lb. per square inch. 

This is a low value even for soft gray iron. The remarkable 
fluctuations in the value of S for specimens of different cross- 
section, from a minimum of 18,700 to a maximum of 36,000, 
show that the ordinary method of calculation would not be of 
much value in predicting the breaking load of such beams. 

Comparison of Strength. — An investigation of the values in 
Table VIII shows that the hollow circular and elliptic sections 
are much stronger than the solid sections, the increase in strength 
being greater than that of the section modulus. The average value 
of S for the last six numbers in Table VIII is 31,600 as against 
24,000 for the six solid sections, an apparent increase in the 
strength of the material itself of over 25 per cent. This is partly 
due to the thinner metal, the greater surface of hard '^skin" 
and the freedom from shrinkage strains. 

The absence of corners and the consequent uniformity of 
metal make this an ideal form of section. 

The hollow rectangles and the I-sections given in Table IX 
have an average value of >S — 22,450. 

No. 8 is lower than the average and Nos. 28 and 32 considerably 
higher. These discrepancies are due to some accidental condi- 



32 



MACHINE DESIGN 



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34 MACHINE DESIGN 

tion of the metal, since the mates of these pieces had about the 
average strength. 

The relatively low values of S for this series are probably due 
to cooling strains in the metal. The table shows quite conclu- 
sively that the increase in strength in such sections is not pro- 
portional to the increase in the section modulus. 

Elasticity. — The values for the modulus of elasticity in Tables 
VIII and IX seem almost ridiculous, if we are to regard this 
much abused '^constant" as any criterion of the stiffness of a 
beam. 

According to the results of tensile and transverse tests on 
cast iron J5' is a variable, being greatest for small loads and 
diminishing as we approach the breaking load. 

Prof. Lanza gives values varying from nine to eighteen millions 
for a test on one bar. As has been explained, the values of E 
were determined from the set lines which were approximately 
straight and not subject to the variation above mentioned. 
Examining the tables we find the values of E ranging aU the way 
from 11,000,000 down to 3,290,000. 

The larger values go with the smaller depths as in Nos. 17 and 
38 and the smaller values are found in the sections having the 
largest section moduli as in Nos. 7 to 11. 

This goes to show that the common formula for E does not 
apply well in the case of cast-iron sections and that the deflection 
of hollow and I-shaped sections is much greater than would be 
given by the formula. The columns giving the loads for a 
deflection of 0.03 in. illustrate this. For instance, the values of 
I for Nos. 1 and 32 are 1.545 and 12.67 respectively, having a 
ratio of 8.2. 

The loads required to produce the same deflection of 0.03 
in. are 2500 lb. and 13,300 lb. respectively, having a ratio of 
only 5.3. 

Rim Sections. — The object of the experiments summarized in 
Tables X and XI was to determine the effect of flanges on the 
strength and stiffness of sections such as are used for the rims 
of fly-wheels. 

In order to illustrate this more clearly each alternate section 
was turned over so as to bring the flanges on the tension side, 
as may be seen by the shapes in the second columns of the tables. 



WHEEL RIMS 



35 



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36 MACHINE DESIGN 

The section modulus and the fiber stress were always calculated 
from the tension side. 

In nearly every instance the calculated value of S is higher 
for the beam having the web in compression and the flanges 
in tension, or in other words there is not so much disadvantage 
in this latter arrangement as theory would indicate. 

For instance, the section modulus for No. 34 is more than 
twice that of No. 13 of similar shape and area, but the breaking 
load is only one-third greater. If we knew where the neutral 
axes of these sections really were during the process of bending 
we might perhaps explain this discrepancy. 

Depth of Flanges. — Another object of these experiments on 
wheel rim sections was to determine the relative value of shallow 
and deep flanges. The average value of the breaking load for 
the ten sections with shallow flanges in Table X is 6690 lb., 
and the average value of S about 25,000 lb. per square inch. 
The corresponding values for the ten sections with deeper flanges 
in Table XI are 11,800 and 22,800. There is thus a slight falling 
off in the value of S for the deeper sections but not so much as 
was noticed in the two other tables. 

The elasticity of these sections is more uniform than in those 
previously noticed, E varying from six to eleven millions. We 
notice, however, the same peculiarity as before, that the deeper 
sections are not so stiff in proportion to the values of / as those 
having shallow flanges. 

The conclusions to be derived from these experiments can be 
stated in a few words: 

(1) The commonly accepted formulas for the strength and 
stiffness of beams do not apply well to cored and ribbed sections 
of cast iron. 

(2) Neither the strength nor the stiffness of a section increases' 
in proportion to the increase in the section modulus or the 
moment of inertia. 

(3) The best way to determine these qualities for a cast-iron 
beam is by experiment with the particular section desired and 
not by reasoning from any other section. 

The experiments described in this article were made with 
unusual care on a remarkably clean and homogeneous iron and 
the regularity of the load curves shows accurate measurement. 



WHEEL RIMS 



37 



That the calculated stresses and moduli show so wide a diver- 
gence must be attributed to the formulas rather than the work 











. 7500 lb. 




No. 33 




/ 








// 


•• 


5000 lb. 






{t 










I 




2500 lb. 




// 


\ 






( 


11 

1 .0 


5 .1 


.1 


5 



Fig. 8. — Load Deflection Curves for Sample No. 33. 

A set of preliminary experiments made on similar sections in 
1901 gave results almost identical with those described, the values 
of S ranging from 22,000 to 35,000 and those of E from five to nine 



38 



MACHINE DESIGN 



millions for a rather hard gray iron. The hollow circular sections 
made the best showing and the thin, deep I-sections the poorest. 



16. Shapes of Frames. — The contours or outlines of machine 
frames vary with the work to be done and the degree of accessi- 
bility desired. They may be roughly classified as follows: 

(a) j-i or parallel type, with symmetrical loading and direct 
tension or compression in parallel members. 

(6) /\ or triangular type, with direct tension or compression in 
inclined members and also in cross girt. 

(c) e; or eccentric type with combined tension and bending in 
long member, bending and shear in two parallel members. 
Similar to the column with eccentric loading. 

(d) C type similar to (c), a semi-circular member being sub- 
stituted for the long straight member. Variable tension and 
bending combined with shear throughout curved part. 

(e) Q or open circular type with variable, combined stresses 
as in circular part of (d). 

if) O ^^ closed circular type with combined stresses varying 
throughout. 

Numerous combinations of these various elements can be 
designed but the principles will remain the same. Table XII is 
convenient for reference. 

TABLE XII 



Type 


Stresses in members 


Illustration 


Vertical 


Horizontal 


(a) H . . 
(6) A... 

(c) E... 

(d) C... 

(e) C . . . 

(/) o... 


Tension, or compres- 
sion. 
Tension, or compres- 
sion. 
Tension and Bending. 

Variable \ 

Combined / 

Variable \ 

Combined f 

Variable \ 

Combined / 


Negligible 

Compression, or ten- 
sion. 
Bending and shear. . . 

Bending and shear. . . 

Variable 1 

Combined J 

Variable \ 

Combined J 


Hydraulic press, slotting ma- 
chine. 
Engine frames. 

Side-crank engine, drill press. 

Punch or shear frame. 

Crane hook. \ 
C-clamp. J 

Chain link. 



Note. — The load is assumed to be vertical in each case. 



FRAMES 39 

17. Stresses in Frames. — The design of frames of the first two 
types in Table XII involves no serious difficulty as the stresses 

are comparatively simple. The ratio - is usually too small to 

permit of buckling in the straight members. As in all cast-iron 
work, care must be taken in proportioning ribs and fillets to avoid 
serious cooling strains and allowance must be made for the 
inferior strength of large castings as compared with small. 

When we consider types (c), (d) and (e) where the loading is 
eccentric and the stresses are composite, the problem is much 
less easy of solution. 

Cast iron is the material most used for machine frames and 
cast iron is not perfectly elastic. The stress-strain diagram is 
not straight but parabolic (see Fig. 8) and presents no well- 
defined elastic limit. 

From Hodgkinson's experiments, the laws governing the 
relations between unit stress and unit deformation were found 
to be approximately expressed thus: 

For tension : 

/S = l,400,000s(l-209s) 
For compression: 

aS = 1,300,000s(1-40s) 
where 

/S = unit stress 

s = unit deformation. 

Since the material does not obey Hooke's law, the ordinary 
formulas for beams will apply only within narrow limits. The 
attempt to apply the more complicated formulas of Resal and 
Andrews-Pearson can only result in a waste of time. ^ 

Under such circumstances, it is best to use simple formulas and 
determine the constants by experiment as has been done in the 
case of columns (see art. 4). 

18. Professor Jenkin's Experiments. — The first, experiments 
so far reported which throw much light on this particular 
problem are those made by Professor A. L. Jenkins and reported 

* For a discussion of these formulas, see Slocum and Hancock's Strength 
of Materials, Chapter IX, and Proc. A. S. M. E., May, 1910. « 



40 



MACHINE DESIGN 



by him in the proceedings of the American Society of Mechanical 
Engineers.^ 

The castings tested by him were eighteen in number and of 
three different forms, all being models on a reduced scale of 
ordinary punch or riveter frames somewhat similar to the one 
shown in Fig. 7. 






Fig. 9. 



Fig. 9 shows the three typical forms chosen : (a) Plain section 
with curved throat; (b) ribbed section with curved throat; (c) 
plain section with straight throat. All of the specimens wer^ 
small, the depth of gap being only 6 or 7 in. 




Table XIII gives the most important results of the experiments. 
The stress in the last column was calculated by the formula, 

S = f + ^ (13) 

the notation being the same as is used elsewhere in this book. 
1 Pbdc. a. S. M. E., May, 1910. 



TESTS OF FRAMES 



41 



TABLE XIII 

Jenkin's Experiments on Riveter Frames 





Strength of 
test bar 


Strength of 
frame section 


Remarks 


Tensile 

stress 


Trans- 
verse 

stress 


Breaking 
load 


Unit 
stress 
(at A) 


1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 


19,100 . 

18,620 

19,000 

21,630 

21,630 

18,600 

18,750 

21,700 

22,920 

20,370 

23,600 

23,000 

24,400 

21,800 

21,400 

21,270 

22,080 

22,800 


36,560 
44,200 
46,080 
37,200 
40,000 
39,000 
43,000 
46,250 
39,600 
43,700 
36,400 
38,000 
45,000 
40,600 
40,400 
37,800 
42,200 
41,300 


11,200 

11,125 

11,390 

9,300 

8,500 

12,600 

12,000 

15,300 

8,300 

8,400 

5,200 

8,400 

5,800 

12,700 

12,500 

11,255 

11,980 

10,600 


16,240 ^ 
16,120 \ 
16,540 J 
11,330 ^ 
10,500 J 
22,520 

9,790 )^ 
12,600 / 
10,1.30 
10,520 
18,420 
10,235 


Same as (a). Fig. 9. 

Same as (6) , Fig. 9. 

(6) with web thickened. 

Tested in compression. 

(6) with outer flange reduced. 
(6) with inner flange reduced. 
{b) with both flanges reduced. 
(b) with both flanges reduced. 
(6) with both flanges notched. 
(6) with fillet strengthened. 
(6) with outer flange removed. 

Same as (c) , Fig. 9. 

Depth of spine reduced. 


23,920 
23,400 
16,320 \ 
17,270 / 
21,476 



That is, the tensile strength at the inside flange is the sum of 
that due to the bending moment and that due to direct tension. 

Some of the different lines of fracture are indicated in Fig. 10, 
the number of each line corresponding to the piece number. 
Number 5 shows an apparent weakness in the web near the 
flange probably due to cooling strains, since the inner flange was 
thicker than the web (see cylinder flanges, pp. 80 and 81). 
Thickening the web as in No. 6 changed this and increased the 
strength (see Table XIII). When a box section is employed, 
the change in thickness between the inner plate and the side plate 
should be gradual. 

Removing or reducing the inner flange always weakened the 
piece (compare Nos. 6 and 10) . Removing the outer flange did 
not always affect the strength (compare Nos. 14 and 15). The 
specimen with a straight throat of the (c) shape usually broke in 



42 MACHINE DESIGN 

the round corner as might be expected from the nature of the 
material (see Art. 12). 

The load-deflection curves obtained in these tests by means 
of an autographic recorder are similar in character to those 
obtained by the author from cast-iron beams (see Fig. 8) and 
show no evidence of a yield-point or an elastic limit. The con- 
clusions reached by Professor Jenkins as a result of these tests are 
' here given verbatim. 

^'Although these experiments are not sufficiently exhaustive 
to render any rigid conclusions, they seem to indicate that the 
following statements are approximately true: 

(a) There is no rational method for predicting the strength of curved 
cast-iron beams suitable for punch and shear frames. 

(b) Of the three formulas suggested for the design of punch frames, the 
well-known beam formula, 

^ My W 

is the most accurate statement of the law of stress relations existing 
in such specimens. 

(c) The stress behind the inner flange at the curved portion is an impor- 
tant consideration that should be recognized by the designer. 

(d) There seems to be no definite relation existing between the strength of 
a curved cast-iron beam and the transverse strength of a test bar cast 
with it. 

(e) The Resal and Pearson- Andrews formulas are unwieldy and awkward 
in their application and offer many chances for error." 

The somewhat erratic variations of the value of the calculated 
unit stresses in Table XIII are rather discouraging to the designer 
but are really no worse than those obtained from simple beams, 
as may be seen by reference to Tables VIII to XI inclusive. 

19. Purdue Tests. — During the past year some experiments on 
curved frames were conducted by Messrs. Charters, Harter and 
Luhn of the senior class in the Testing Materials Laboratory of 
Purdue University. 

The characteristic shape and dimensions of the specimens are 
indicated in Fig. 11, while Fig. 12 shows the piece in position in 
the testing machine. The load was applied by means of stirrups 
carrying round steel pins which bore in the milled grooves shown 
at G, Fig. 11. 



RIVETER FRAMES 43 

The proportions of the frame were copied from those of a large 
hydraulic riveter made by a reputable firm. The castings were 
of a uniform quality of soft gray iron and were made in the 
university foundry. 

Test pieces for tension and flexure were cast from the same heat 
and showed an average tensile strength of about 25,000 lb. and a 
modulus of rupture of about 41,500 lb. 

Twenty-four pieces were broken, the same pattern being used 
throughout, various modifications being made in the flanges and 
fillets. 



Fig. 11. 

The following table shows these modifications in detail and the 
effect which they had on the strength and stiffness of the frames. 
Some of the characteristic lines of fracture are shown in Fig. 11, 
each line being numbered to correspond to the number of the 
specimen. 

The first twelve specimens broke by splitting the web along a 
curved line parallel to and adjacent to the inner flange. This 
type of break has already been discussed in Art. 18. 

Numbers 13 to 16, inclusive, broke directly across the frames 
in lines parallel to one of the radial ribs. 

Numbers 17 and 18 broke in much the same manner in lines 
parallel to the one rib. 

Numbers 19 and 20 started a fracture in the web adjacent to 
the rib but this did not extend through the flanges. 

The last four frames broke in a practically vertical line through 
web and flanges just back of the inner flanges. 



44 



MACHINE DESIGN 



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RIVETER FRAMES 



45 



A study of the values given in the table and of the lines of 
fracture in Fig. 11 shows the difficulty of applying any general 
formula to this problem. 

The general tendency of all the frames which are not reinforced 
by radial ribs is to split or shear in the web. Probably all of 




Fig. 12. 



the fractures begin in this way and it is more or less a matter of 
chance whether the fracture extends through the flanges. 

When ribs are used, the tendency is still to shear the web 
alongside of the rib as in 16 and 18, with a possibility of the break 
not extending through the flanges (see No. 20). 

It is apparent that thickening the flanges will do no good, 
while thickening the web is efficient (see Nos. 21-22). Changing 
the thickness of web from f in. to 0.6 in. increased both strength 
and stiffness nearly 100 per cent. 



46 



MACHINE DESIGN 







DESIGN OF FRAMES 47 

The addition of ^-in. fillets increased strength and stiffness, 
while J-in. fillets were less effective. 

Although these experiments were not sufficient in number to 
justify definite conclusions, it is evident that the web and not 
the flanges is the weak part of the ordinary G frame and that 
reinforcement of this by increasing its thickness or by the addition 
of radial ribs is the rational method of treatment. 

It is also evident that the experiments just quoted substanti- 
ate many of the conclusions reached by Professor Jenkins. 

The application of the common formulas for beams to the 
results given in Table XIV gave values for the unit stress which 
are contradictory and misleading. The more complicated formula 
of Bach^ was equally unsatisfactory. 

Further experiments may lead to empirical formulas, which will 
answer for all ordinary purposes of design. 

20. Principles of Design. — In designing a frame for a punch or 
shear press similar to those shown in Figs. 7 and 9, attention must 
be paid to the stiffness as well as the strength, since any sensible 
deflection or distortion will cause trouble with the dies and 
punches which do the cutting. In future experiments, it is de- 
sirable that careful attention be paid to the relative stiffness of 
various sections. It is probable that the thickness of the web 
and the weight of the outside flange have much to do with stiff 
ness and that these have sometimes been neglected when strength 
alone has been considered. 

Formula (13) may be put in the following shape for convenient 
use: 

S 



w= 

ly I (14) 

7^1 



where 



TF = pressure between dies. 
S =safe tensile stress on material. 
I = perpendicular distance from line of pressure to 
neutral axis of section. 

- = section modulus (tension side). 

y 

Bach's Elasticital and Festigheit. 



48 MACHINE DESIGN 

A =area of section. 
The formula applies to any horizontal section as AB, Fig. 10. 
For any inclined section, the equation becomes: 

S 



w 



ly cos a (14a) 



I ' A 

where 

a = angle made by the section with the horizontal. 

For any section parallel to the line of pressure, the second 
term in the denominator disappears and the formula is the same 
as for an ordinary cantilever beam. 

The stress in the web at any section in the curved spine of the 
frame is largely tension and may account in part for a fracture 
like No. 5 in Fig. 10. 

This may be illustrated in Fig. 10 by considering the outer and 
inner flanges as separate members connected by radial lattice 
work. It is evident that pressure tending to open the gap 
would also tend to move the flanges further apart, increase the 
distance AB and subject the radial lattice bars to tension. To 
meet this condition, some manufacturers introduce radial ribs as 
shown in Fig. 7.^ Some manufacturers provide means for rein- 
forcing the gap in a shear or punch frame by steel stays which can 
be attached when especially heavy work is to be done. Fig. 13 
illustrates a frame of this character. The machine has a 60-in. 
gap and is capable of punching a 2|-in. hole in 1^-in. iron or 
shearing a bar of flat iron 1|X8 in. 

There is always present the possibility that the neutral axis of 
any section does not exactly coincide with its center of gravity, 
especially in the curved portion, but the uncertainties of the 
material itself outweigh any consideration of this sort. 

Straight Frames. — ^Frames which have a straight spine like 
those of drill presses, slotting machines and profiling machines, 
are similar in condition to type (c). Fig. 9, and have a uniform 
bending moment in the straight part combined with uniform 
tension. The condition is that of a column in tension with 
eccentric loading and the deflection is usually the thing to be 
considered rather than the strength. This may be illustrated by 
the ordinary iron clamp such as is used in foundries and 

^ For further discussion of this point, see.Professor Jenkin's paper. 



DESIGN OF FRAMES 49 

pattern shops and which sometimes assumes the shape shown 
in Fig. 14. Practically the frame is more likely to break at the 
curved portion joining the column or spine with the horizontal 
members. This is doubtless due to the shrinkage strains caused 
by the profile at this point. 



The frame of a side-crank engine is a good example of the 
straight frame with eccentric loading. The points of rupture are 
apt to be at the junction of the frame with the cylinder flange or 
near the main bearing. 

REFERENCES 

Modern American Machine Tools. Benjamin. 

Evolution of Machine Tools. Cass., Dec, 1898; Cass., Sept., 1904. 

Things that are Usually Wrong. Am. Mach., Mar. 9, 1905. 

Design of Boring Mill. Am. Mach., Mar. 8, 1906. 

Cast Iron in Machine Frames. Am. Mach., Oct. 24, 1907. 

Design of Machine Frames. Mchy., Aug., 1908. 



CHAPTER III 



CYLINDERS AND PIPES 



21. Thin Shells. — Let Fig. 15 represent a section of a thin shell, 
like a boiler shell, exposed to an internal pressure of p pounds per 
square inch. Then, if we consider any diameter AB, the total 
upward pressure on the upper half of the shell will balance the 
total downward pressure on the lower half and tend to sepa- 
rate the shell at A and B by tension. 




Fig. 15. 



Let - d = diameter of shell in inches 

r= radius of shell in inches 
I = length of shell in inches 
t= thickness of shell in inches 
S= tensile strength of material. 

Draw the radial line CP to represent the pressure on the element 
P of the surface. 

Area of element at P = Irdd. 
Total pressure on element = plrdd. 
Vertical pressure on element = plr sin ddO. 

Total vertical pressure onAPB = I plr sin ddd =2plr. 

50 



THIN SHELLS 51 

The area to resist tension at A and B = 2tl and its total strength 
= 2tlS. 
Equating the pressure and the resistance 

2tlS = 2plr 
pr_pd 
^~S~2S ^^^^ 

The total pressure on the end of a closed cylindrical shell = 
Tzr^p and the resistance of the circular ring of metal which resists 
this pressure = 2nrtS. 

Equating: 2KrSt = 7zr^p 

Therefore a shell is twice as strong in this direction as in the 
other. Notice that this same formula would apply to spherical 
shells. 

In calculating the pressure due to a head of water equal h, 
the following formula is useful: 

p = 0.434/1 (17) 

In this formula h is in feet and p in pounds per square inch. 

PROBLEMS 

1. A cast-iron water pipe is 10 in. in internal diameter and the metal is 
f in. thick. What would be the factor of safety, with an internal pressure 
due to a head of water of 250 ft.? 

2. What would be the stress caused by bending due to weight, if the pipe 
in Ex. 1 were full of water and 24 ft. long, the ends being merely supported? 

3. A standard lap-welded steam pipe, 6 in. in nominal diameter is 0.28 
in. thick and is tested with an internal pressure of 500 lb. per square inch. 
What is the bursting pressure and what is the factor of safety above the test 
pressure, assuming *S = 40,000? 

22. Thick Shells. — There are several formulas for thick cylin- 
ders and no one of them is entirely satisfactory. It is, however, 
generally admitted that the tensile stress caused by internal 
pressure in such a cylinder is greatest at the inner circumference 
and diminishes according to some law from there to the exterior 
of the shell. This law of variation is expressed differently in the 
different formulas. 

Barlow's Formulas. — Here the cylinder diameters are assumed 



52 MACHINE DESIGN 

to increase under the pressure, but in such a way that the volume 
of metal remains constant. Experiment has proved that in 
extreme cases this last assumption is incorrect. Within the 
limits of ordinary practice it is, however, approximately true. 
Let rfj and d^ be the interior and exterior diameters in inches 

and let t= ^ q ~ be the thickness of metal. 

Let I be the length of cylinder in inches. 

Let iSj and S2 be the tensile stresses in pounds per square inch 
at inner and outer circumferences. 

The volume of the ring of metal before the pressure is applied 
will be: 

and if the two diameters are assumed to increase the amounts Xj 
and X2 under pressure the final volume will be: 

Assuming the volume to remain the same: 

d^^ — d^^ — (^2 + X2) ^— {d^ + xj ^ 
Neglecting the squares of x^ and x^ this reduces to: 

or the distortions are inversely as the diameters. 
The unit deformations will be proportional to 

-^ and -J 

and the stresses S^ and S2 will be in the same ratio: 



A3 -I »//-| Ct'2 ^' 



or the stresses vary inversely as the squares of the diameters. 
Let S be the stress at any diameter d, then: 

S d ^ S r ^ 
S= \2^ = %^ (where r is radius) 

and the total stress on an element of the area l.dr is: 



THICK SHELLS 



53 



Integrating this expression between the limits ~ and -^ for r and 
multiplying by 2 we have: 



^-2s.v<|-|>2«''<^- 



(h) 



Equating this to the pressure which tends to produce rupture, 
pdl, where p is the internal unit pressure, there results: 



P 



2S,t 



d^ + 2t 



(18) 



The formula (15) for thin shells gives p = — 



2St 



d 



By comparing this with formula (18) it will be seen that in 
designing thick shells the external diameter determines the work- 
ing pressure or: 

2S,t 



V 



dn 



(18a) 



Lame's Formula. — In this discussion each particle of the 
metal is supposed to be subjected to radial compression and to 




Fig. 16. 

tangential and longitudinal tension and to be in equilibrium 
under these stresses. 

Using the same notation as in previous formula: 



for the maximum stress at the interior, and 



(19) 



54 MACHINE DESIGN 

0/7 2 

S. = ,|^P. (20) 

for the stress at the outer surface. 

Fig. 16 illustrates the variation in S from inner to outer 
surface. 

Solving for d^ in (19) we have 

d, = dJ^±^- (21) 

A discussion of Lame's formula may be found in most works on 
strength of materials. 

PROBLEMS 

1. A hydraulic cylinder has an inner diameter of 12 in., a thickness of 4 
in. and an internal pressure of 1500 lb. per square inch. Determine the 
maximum stress on the metal by Barlow's and Lame's formulas. 

2. Design a cast-iron cylinder 8 in. internal diameter to carry a working 
pressure of 1200 lb. per square inch with a factor of safety of 10. 

3. A cast-iron water pipe is 1 in. thick and 18 in. internal diameter. 
Required head of water which it will carry with a factor of safety of 6. 

23. Steel and Wrought-iron Pipe. — Pipe for the transmission 
of steam, gas or water may be made of wrought iron or steel. 
Cast iron is used for water mains to a certain extent, but its use 
for either steam or gas has been mostly abandoned. The weight 
of cast-iron pipe and its unreliability forbid its use for high 
pressure work. 

Wrought-iron pipe up to and including 1 in. in diameter is 
usually butt-welded, and above that is lap-welded. Steel pipes 
may be either welded or may be drawn without any seam. 
Electric welding has been successfully applied to all kinds of steel 
tubing, both for transmitting fluids and for boiler tubes. 

The tables on pp. 56 to 61 are taken by permission from the 
catalogue of the Crane Company and show the standard dimen- 
sions for steam pipe and for boiler tubes. 

Ordinary standard pipe is used for pressures not exceeding 
100 lb. per square inch, extra strong pipe for the pressures pre- 
vailing in steam plants where compound and triple expansion 
engines are used, while the double extra is employed in hydrau- 
lic work under the heavy pressures peculiar to that sort of 
transmission. 



BOILER TUBES 55 

Tests made by the Crane Company on ordinary commercial 
pipe such as is listed in Table XV showed the following pressures: 

8 in. diam 2,000 lb. per square inch. 

10 in. diam 2,300 lb. per square inch. 

12 in. diam 1,500 lb. per square inch. 

The pipe was not ruptured at these pressures. 

24. Strength of Boiler Tubes. — When tubes are used in a so- 
called fire-tube boiler with the gas inside and the water outside, 
they are exposed to a collapsing pressure. 

The same is true of the furnace flues of internally fired boilers. 
Such a member is in unstable equilibrium and it is difficult to 
predict just when failure will occur. 

Experiments on small wrought-iron tubes have shown the 
collapsing pressure to be about 80 per cent of the bursting 
pressure. With short tubes set in tube sheets the length would 
have considerable influence on the strength, but ordinary boiler 
tubes collapsing at the middle of the length would not be in- 
fluenced by the setting. 

The strength of such tubes is proportional to some function of 

-, where t is the thickness and d is the diameter. The formulas 
a 

heretofore in use are very limited in their application, being 
founded on experiments covering but a few diameters and 
thicknesses. 

Fairbairn's formula is the oldest and best known of these and 
was established by him as a result of experiments on wrought- 
iron flues not over 5 ft. in length and having relatively thin 
walls. 

f 2.19 

p = 9, 672,000-^^-^ (22) 

all dimensions being in inches and p being the collapsing pressure. 
D. K. Clark gives for large iron flues the following formula: 

P = 20^0^ (23) 

where P is the collapsing pressure in pounds per square inch. 
These flues had diameters varying from 30 in. to 50 in. and thick- 
ness of metal from | in. to y^g in. 



56 



MACHINE DESIGN 



I— ( 

Ph 



< o 

h^ ^ T5 

o 

§^ 

M 

H 

a 
a 

o 



Number of 

Threads 

per inch of 

Screw. 




00 
1— 1 


00 
1—1 




1—1 


1—1 


T—l 


■I— 1 


1—1 


00 


00 


a; 

111 


a 
a 



1— ( 
0? 


1 05 
1 -* 


05 


CO 
00 


T— I 

1— I 


00 
CO 

7—1 


03' 


00 

CO 


05 



CO 
CO 


Ci 
CO 

»o 


CO 

CO 

10 


Length of 

Pipe 
Containing 

One 
Cubic Foot. 




CO 

10 

C3 


CO 

CO 
00 
CO 

•r— ( 


T-l 





03 


05 


CO 

1—1 


10 

CO 

05 


CO 

CO 
d 


1—1 
Ci 


d 

CO 


10 

ci 








at 

o m 


M 

a a 


0) 


10 

tH 
tH 


05 

^. 


1—1 


co 


CO 
1—1 

z6 


10 

CO 
CO 


10 


co' 


OD 
CO 


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CO 


00 

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JO 

03 


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t< 3 

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10 



10 


10 


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co" 




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00 





GO 




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GO 
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1—1 


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a; 
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a 

72 





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tH 


CO 



1— 1 




CO 

00 


^ 


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00 

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00 




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1 




CO 

10 




1—1 


1—1 



CO 


CO 
CO 
CO 


CO 

03 CO 
CO Oi 
00 XH 


00 

CO 




CO 

iO 
CO 

CO 


00 


00 
00 
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■3 

a 

0) 





1— 1 


05 


QO 
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00 


00 
JO 
CO 

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to 

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00 




Ci 

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s 

3 
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00 


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03 

10 

T-H 


tH 


05 
00 

to 

C5 


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CO 


JO 
CO 
CO 


T—l 

CO 


to 


Ci 
CO 


CO 
JO 


CO 

CO 

ci 


"3 
5 


a 


01 

01 





55 

o5 


CI 
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05 

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10 


Ci 

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Ci 


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Ci 
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d 




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2 - =1^ 
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a 


00 




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Ci 



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CO 
tH 


1—1 


tvO 

1—1 


iO 

1— ( 



c? 


T-l 




1 

5 


Approxi- 
mate 
Internal 
Diameter. 



.a 

a 

h- ( 




CO 


OS 


CO 




00 


00 



1— 1 


00 
CO 

T—l 


1-1 
1—1 
CO 
tH 


CO 



00 

CO 


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a) 
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a 
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00 








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a 


rHlQO 


^k 


coix <-^ 


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1—1 


0--? 




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STEAM PIPE 



57 



o 



> 

X 
w 

PQ 
< 



P-H 

Ph 



<! 

02 

o 



o 
§ S, 

« 

M 
I 

K 
o 

o 



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Threads 

per Inch of 

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00 


00 


00 


QO 


GO 


00 


00 


00 


00 


30 


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1 


1—1 



05 


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tH 


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tH 

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00 




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CO 


10 

CO 

CD 

d 


GO 

o> 



id 


10 

00 
OS 

00' 


Length of 

Pipe 
Containing 

One 
Cubic Foot. 




1— 1 


T-( 

CO 
1— 1 


o-j 


OS 




00 

OS 


CO 


00 
00 

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01 

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00 

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Is 


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05 


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1 
i 

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1 


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£3 




T-H 

co' 


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T-l 

CO 


00 


10 


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OS 
CO 


CO 

00 
CO 

00 


CO 


d 


OS 

T-l 


T-l 



CO 

tH 


OS 
i> 

10 

■r-H 


g 


t— t 


s 


00 
00 

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CO 
tH 


■1-H 
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OS 

10 
1—1 


OS 

OS 

C5 

T-l 


00 

00 

00 
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<>1 


GO 

CO 

J> 

00 

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10 


CO 

07 

CO 


OS 

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GO 

00 


CO 

CO 



uo 

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00 

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CO 
tH 

tH 


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c 

l-H 

02 


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10 

T-l 


10 

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CO 


CO 


-* 

CO 


CO 
CO 

10* 


CO 
GO 

to 


CO 


CO 
CO 

d 

OS 


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00 



T-l 


CO 
0? 




aJ 
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a 

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5 


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g 

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c 




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00 


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00 
10 





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10 


CO 



00 


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00 

10 
10 

^' 

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10 




00 



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tH 


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tH 

00 

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OS 

CO 
C5 


CO 

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07 


QO 
CO 

0^ 

0* 

CO 


0? 

£> 

CO* 
CO 


OS 

d 
CO 


10 

iO 


d 

Tf 




Nominal 
Thick- 
ness. 

Inches. 


07 


co 
(?■» 


CO 


OS 




00 

0? 


T-l 



CO 


0? 

CO 


CO 


CO 
CO 
CO 








<D 

i 

5 


Approxi- 
mate 
Internal 
Diameter. 


01 



00 

10 

CO 


03 




00 


to 


10 



iO 


CO 



CO 


CO 




C3 
00 
OS 


CO 
OS 

GO* 


OS 

•r—i 


d 

tH 


T-l 


0? 

T-l 


-2*3 

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a> 

ja 


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in 

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d 

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a 
p-i 


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to 


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t- 


00 


OS 




T-l 




CI 

T-l 



58 



MACHINE DESIGN 



I— I 
PL| 

o 

!? 

o 
r^ .2 

« 2 

'^ -5 

ffl «^ § 

H !^ ^ 
O 

W 
o 
p 
o 
« 






Nominal 

Weight 

per Foot. 


a 

3 


Ah 


<35 
0* 


^ 


^^ 





OS 

l-H 


2 


CO 


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CD 

CO 


g 



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in 









s 
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0? 
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to 


CD 





eo 


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in 
m 


in 

l-H 


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l-H 


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l-H 
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c6 oj 
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a 

03 

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as 



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OS 

l-H 


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1 
in 


a 

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to 

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l-H 


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CD 


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CM 

a 

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a 

0) 

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l-H 


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;=: 


3 

l-H 


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05 




CO 


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lO 


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in 


l-H 

CO 

1* 


OS 


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d 


>n 


g 




CO 




Nearest 

Wire 

Gauge. 


d 




l-H 
T-H 




l-H 


OJ 


00 


t- 


CD 


CD 


in 


0« 


- 














8 


Nominal 
Thick- 
ness. 


a 

l-H 


- 


SO 

7i 


IM 


OS 

l-H 


tH 


1 


1 




5* 


^ 


eo 


CO 


tH 

SO 


in 
CO 


S 
-* 




u 
a) 

s 
5 


<3 1-1 


CO 

.3 

3 

1— 1 


tn 


^ 
« 


•<* 


04 

»o 




i-< 

05 


?2 

0« 

l-H 


S 
^ 


CO 


in 
eo 


00 


CO 

so 


00 
00 
so 


CO 

00 


in 
in 




to 

<D 

.3 

3 

H- 1 


to 




•* 





^ 


^ 


in 

l-H 

CO 


l-H 


l-H 


eo 


J2 

00 


in 


■>* 


•n 


in 


in 
<>« 

CD 
CO 


11 

a^ 


to 

0) 

J3 


a 

hH 


^ 


^ 


^ 


:^ 


^ 


- 


;5t 


^ 


N 




CO 


so 


■<t 


• 
in 


CO 



^3 
(V 



(V 



I— ( 

a. 
3! 
o 
O 

o 

w 

73 

ca 
a; 

fH 

H 

+3 
3i 
o 



73 

IB 



0) 

a 

bfi 
CI 
O 



m 



H-3 



HYDRAULIC PIPE 



59 



XI 

PQ 
< 



H 




fLi 




1— ( 




fin 




O 




^ 




O 




tf 




H 




!7J 






rn 


<! 


fl 


rt 


O 


H 


• r-l 


>< 


ri 


w 


o 


H 


a 


.J 


Q 


P 


Tl 


o 


Ui 


Q 


c3 


h:) 


fl 


1^ 


03 






C/J 


«4-l 




o 


Q 




^ 


a; 


<! 


^ 




(rt 


O 


H 


w 




1— 1 




H 




W 




o 




p 




o 




w 




i^ 





1^1 



p-<y 






O 



03 <0 

as 



hHCO 



c3 ® 






C3 I 



CO 

-a 


i> 


^ 


« 

» 


<M 


■* 


§ 


§ 


s 


^ 


00 


N 


o 


'-' 


OJ 


eo 


in 


CD 


05 


00 


00 






CO 


PL| 

























i-H Tr 00 





I> 


as 


rr) 


r^ 






to 


(N 


o 


r- 


o 




eo 


■^ 


o 




1-H 


o 


<^ 


i~ 


o 




Tt< 




CO 


o 


Hi 


so 


o 


iO 




o 


T)< 


w 


05 


U5 


OS 


CO 


■* 


eo 


03 


OJ 


■^ 


'"' 


■^ 





, 


t- 


r- 


Tf 


s 




on 


on 




»n 


05 


r^ 


0) 


■>* 


eo 


o 


l-H 




(V 


C5 


>o 


Tt 




»o 


!0 


O) 


eo 


O 


CO 


eo 


o 


05 


00 


CO 


fe 


-* 


eo 


« 


(M 


(M 


tH 


l-H 


1-t 











^ 


r- 


f^ 


35 


.O 


CD 


eo 


•^ 


c» 






o 


(^» 


cn 


Tf< 




00 


^^ 


c» 


f^ 


00 


Tf 


^■p, 


in 


t- 


o 


iCi 


Oi 


CD 


o 


in 


t- 




eo 


1— I 






T-l 


tH 


TH 


0« 


Tt< 


in 


co 


00 


^ 



. 0) 


t- 


05 


rH 


o 




Tl< 


!T5 


f^ 


-^ 


■^ 


Tf 


CO 


f- 




CO 


Tf 




CT> 


05 


OJ 


cr^ 


O 


i-c 


(^i 


CO 


05 


t^ 


Ttl 


O 


1^ 


f- 


wo 












t-H 


03 


■^ 


in 


!> 


t-i 























in CO 

CO CO 
OVCO 



03 


-* 


CO 


on 


^ 


lO 




C3 




cn 


•^ 


CO 




in 


CO 


in 


CO 


CO 


eo 


CT5 


OJ 


CO 


<-) 


o 


7-?. 


in 


00 


eo 




00 


Tf 


TJ< 




in 


G5 


eo 


ITl u 






.,_, 


0? 


03 


rf 


CO 


05 


w 


lO 


■^ 


1— 1 




















" 


03 



W 


CO 


CO 


Tf 




on 


-^ 


CO 


ir 


eo 


03 


Tt< 




CO 


03 


-* 


Of) 




f/) 




1^ 


CO 


in 


CD 


A 


J> 


CO 


(JO 


t- 


-^ 


CO 


in 




in 


00 


t- 


a 




•^ 


--l 


03 


CO 


Tt< 


in 


i> 


00 


CJS 


03 



























CO 


C5 


05 




in 


05 


T-l 


03 


CD 


CO 


r- i> 




CO 


CS 


eo 




CD 


CD 


CO 


05 


CD 


eo t- 


A 


CO 


03 




03 


OS 


TT 


o 


Ci 


in 


»-< ^ 


o 


03 


eo 


Tt< 


in 


in 


r- 


C5 


o 


03 


TP £>^ 













































th ,-1 T-1 1-1 03 

















1 
1 


1 


X 


1 




d 










o 


o 


CO 

TH 


i?? 


is 


x> 


;;^ 






o 


o 




o 


— 






--- 






'"' 


'^ 


o 


o 


'^ 


'■"' 








-^ 





t— I r<Vs fSs 



EC 

ID 


00 


Tt 


^ 


Tn 


;r) 


03 


O 


GO 


03 


03 




05 




» 


Of) 


o 


^ 


-X) 


O 


Tf 


QT) 


in 


A 


03 


eo 


CO 


00 


Tf 


■* 


in 


CD 


CO 


CD 


i- 


o 
























a 
























1— I 

























^ (B «e • . 

tH eg tl g (D 



W_ 

"3 IS 

a a 

^5 



•^ 


03 


r^ 


lO 


30 




lO 


Tl< 


CO 


•£> 


00 


T)< 


03 


.-n 


OT) 




05 


in 


00 




CO 


CO 


03 


-<* 


in 


00 


'-' 


■^ 




03 


t^ 




o 










T-H 


'"' 


'"' 


03 


03 


CO 


Tf 



^ 



;^ 



::^ 



:^ 



:^ 



::?! 



73 



o 



to 

'a 

o 
O 



T3 

(^ 

+=> 
;=! 
o 



T3 



O) 

bC 
i=l 
O 
;h 

02 



+3 



O 



60 



MACHINE DESIGN 





r/J 






W 






W 






P 






H 






« 






Px5 






h) 












o 


t/J 

o 




o 


<1) 


hH 


l-H 


s 


l-H 


iJ 


« 


> 

X 


<1 

o 
o 


-73 

n1 


W 




7:3 

"1 


H^ 


w 


w 


o 


-M 


-^ 


Ph 


UJ 


H 


o 


o 




J 


T) 




M 


1— 1 




H 


ri^ 




H 


a 




O} 


H 




Q 






W 













h-l 






w 






^ 






PM 






-^ 






)-\ 





Nominal 

Weight 

per 

Foot. 


a 
fl 
o 


o 

05 


1— ( 


o 


CO 

CD 

T-l 


T— 1 


CO 

T— 1 

oi 


lO^ 


o 
co' 


CO 
CO 

CO 


CO 

OS 
CO 


00 
05 

-** 




lO 


T-H 
CD* 


00 
iO 




u 

tJOC 
k5 


a o 

(D =1-1 
-(J t-c 

•-ICC! 


CD 


07 

CO 


CO 

iO 

CO 


CO 
CO 
00 


X) 
05* 


1—1 

1-H 

C5 


)0 
00 

l-H 


CD 
T-H 


05 

o 

IO 

1-H 


CO 
00 

T-H 


CO 
05 


o? 

1-H 
T-H 


00 

oo 
o 

1— i 


o 

tH 


o? 

o 
OS 


C3 

T-H 

00 


"5 a5 




05 

T— 1 

00 
CO 


CO 
IO 

o 

CO 


c- jco 

-* loo 

C7 c<i 


as 00 

O Oi 

1—1 1-H 


00 

o> 

lO 


05 CO 
00 i> 
CO OJ 

1—1 pH 


iO 

T— 1 

tH 


tH 
OS 

o 


00 
tH 

o 
1—1 


iO 
OS 


OS 
-<* 

00 


CO 




03 

c3 
u 

<v 

03 
^< 

> 

C 
cS 
u 
H 


|a2| 


CO Oi 'dH 

CO CO Oi 

O^ iCO -* 


05 
CO 
lO 


CO 
CD 


05 

•r-^ 

00 


IO 

o 

• 


05 


1 

00 {> 
1-1 C5 

tH -pH 


OS 
CO 
00 


t> 00 CD 
05 CO |iO 
CO 00 |05 

l-H T^ d 


IS 

u 

fl 

1— 1 


.fl 

a 

fl 

l-H 




1—1 
CO 


00 tH 

CO Oi 


CO 


CO 
CO 
00 

oo' 


o 

rl^' 


IO 
Ot) 

o 
o 


OS CO 

t- 1-H 

O 1-H 

CO li> 


CO 
00 


CO 

CD 
OS* 


OS 

CO 

crs 
d 

tH 


CO 

CO 
o 

^' 

T-( 


OS 

00 
1-H 


o 


.fl' 
c 

cr 

CO 


00 




t- liO 

CO o 

t- I-* 


(» CD 

1-1 05 
co' CO 


o 
Oi 

^' 


OS 

IO 


OS 
CO 

o 


CD 

OS 

o? 

00 


1—1 
CO 
OS 


iO CD 
^ CD 
O IO 

l-H lo? 
1-H tH 


o 
OS 

lO 

T-H 


IO 
CO. 
CD 

os' 




o 

a 

a 

o 

5 


la 
fl 

Ol 

fl 


.fl 

o 

fl 

1— 1 


00 

CO 


CO 


T-H 


1—1 

o 


CO 

00 

CO 

IO 


CO* 


CO 

1-H 


IO 


00 


CO 
IO 

os' 


T-H 

o 

1-H 


o? 

tH 

1-H 


'^ IO 

05 'os 

tH CO 

1-H T— 1 


00 

r-H 


"5 
fl 
U 


o 

a 
t— 1 


T-t 

CO 


co' 


pH 


00 
Oi 

lo' 


00 
00 

co' 


91 
o 


^ Oi IO 
IO CO o> 
oo CD xf< 

t- 00 Oi 


T-H 

o 

T-l 


CO 

OS 
OS 

o 

l-H 


T-H 

00 

T-H 


CD 
CD 
IO 


CO 
T-H 


00 

o 

iO 


^1 


6 


CO 


CO 
1—1 


CO 

T— 1 


CO 

l-H 


CO 
1-H 


CO 

T-H 


03 
1-H 


1— 1 


05 

1-1 


1-1 

1-H 


tH 
r-i 


T-H 


o 


o 

1-H 


OS 


Nominal 
Thick- 
ness. 


o 

fl 


Iff 
o 


o 


»o 

CD 

o 


JO 

o 


IO 

o 


LO 

Oi 

o 


o 

1-H 


05 

o 

1-H 


1 

OS i 

O 05 
tH T-H 


C5 

^H 


CO 

1-H 


CO 
tH 




1-H 




u 

s 

5 


fl 

fl 


03 
.fl 

fl 


CO 

QO 


CO 

o 

T-H 


CO 
CO 

1—1 


CO 

iO 


1— 1 
00 

T— 1 


CO 

o 


00 

o? 
oi 


C5 

CO 

iO 
OJ 


05 
GO 

od 


^ CO 
O Oi 

CO CO 


1-H 

IO 
CO* 


C5 

CO 

CO 


c. 

CO 


o 


"3 
fl 


03 
0) 

o 

fl 


-pH 


Hi* 


1—1 T-H 


03 


03 


C3 


03 


CO 


1 

Hrj. 

CO 


CO 


ecHi 
00 


rl^ 


T*' 

^ 


iO 



BOILER TUBES 



61 






X 

PQ 






w 






O 


^ 


rr 


o 


iH 


« 


O 


1— 1 


a 


h3 


•1-1 


o 
o 


-H 


« 


cfl 


-»1 


-^3 


w 


i=l 


C ) 


a 




■*-> 


« 


m 


o 


Cm 




o 


1-3 


<i) 


N 




;^ 


^ 


H 


c3 


OQ 


H 



Q 
I 



<«5 . 
1^^ 



73 

C 


CD 




IC 


;o 




CO 


CD 


CD 




r- 1 


C5 


o 


c- 




o 


^ 


O 




c 


o 


1— ( 


CO 


CD 


1-H 


lO 


on 


o> 


CD 


■»— ( 


I— 1 




1—1 


Oi 


Cv^ 


o< 


CO 


CO 



«^ 

S 3 

bocr 
ceo 

h4 




CO ,CO 


00 


o 


00 o? 




iO 


CO 


t- It- 


Oi 


■^ 


05 CD 


CO 


o 


on 


CD IC 


^ 


^ 


CO 1 CO 


CO 


00 


CJ 







C9 c- 

00 ^* 
CO CO 



-^ ico 

Oi t- 



■2 \B 

en 1 1-1 



o 



05 



^1 



03 1 






lO 


CO 


o. 


on 


O) 




05 


CO 


o> 


-* 


CD 


00 


-rt* 


lO 


CD 


^ 


o 


»o 


o 


OS 


(>< 


^ 


^ 


la 


CO 


CO 


TJH 


^ 


CO 


«> 


00 


05 



t- lO 



O |Tt< 






1 <^ 


o 


'^ 


05 


IC-"? 


CO 


OS 




l-t* 


-* ^ 


o 


(» 


Ci 00 


o> 


31 


lo> 


c? 


05 


c^ 


CD 


Ic^? 


o 


CD 


T— ( 


\oi 


la 


^ 


CD 


00 


c? 


r^ 


^ 


CO 


Ico 


C3 


CO 


Tt^ 


iO 


f> 


00 


jO 


r-H 


■I— 1 



-^ lO 
}> 00 
CM i-^ 



00 00 
C? CO 



CD lO 



CO lOO iCO 
CO C5 CO 

o o U> 



CO lOl 

tH ico 



00 I 
CO CD 
05 O 



CO -^H 



CO 


^ 


CD 


00 




o 




iO 


^ 


tH 


O 


05 


^ 


-* 


/^ 


CD 


^ 


o> 


00 


05 


1"^ 


1— 1 


^ 


rH 


1^ 


CO 


-^ 


r- 


o 


•rt< 


r^ 


o 


CO 


CD 


OS 


0-) 


T-( 


o< 


C^ 


ot 


CO 


CO 


CO 


CO 


^ 





T-H 


CO 


^ 


CD 


IC 


OS 


CO 


t- 


tH 


oo 


OS 


T-H 


Of 


^ 


00 


■I— 1 


lO 


T) 


•rH 


T-l 


Ol 


o, 


c^ 


CO 



OS 


-l-H 


Cv> 


OS 


^ 


OO 


CD 


GO 


OS 


r- 


O 


CO 


CO 


Tj< 


"* 



00 30 lOO 



CD »o r^ 



I ■-tn I y-tpi 

T^ |CO |W 



33 


ic 


l-C 


lO 




CO 




OS 


00 00 




CD 


CD 


CD 


00 


o 


Oi 


(M :CO 1^ 1 


o 


T-l 


T-H 


1—1 


T-H 


Oi 


Oi 


o? 


CQ IC5 


c 


• 








" 






• 1 • 


1— ( 
















1 



t- ^■ 



rH OS CD 
CD UO lO 



00 lOS 



<M rt^ r^i Gv> 

-^ 0-> O CO 
lO lO i-O rjt 



0) 

o 


CD 


t- 


00 


OS 


o 

i-H 


T-H 


C5 

T-H 


CO 

T-H 


i 

rh 

irH 



62 MACHINE DESIGN 

In 1906, Professor R. T. Stewart reported to the American 
Society of Mechanical Engineers some very comprehensive 
and interesting experiments on lap-welded boiler tubes of 
Bessemer steel. ^ 

The tests were conducted at the works of the National Tube 
Company on tubes manufactured by that firm and were in 
progress for four years. 

Two series of experiments were made — one on tubes 8f in. 
outside diameter of different thicknesses and of different lengths, 
for the purpose of testing the applicability of existing formulas 
to tubes of this character; one on tubes 20 ft. long and of different 
diameters and thicknesses for the purpose of establishing empirical 
formulas for the strength of such tubes. 

The formulas of Fairbairn, Clark, Unwin, Grashof, etc., were 
tested by comparison with the results of the first series of experi- 
ments and were all found inapplicable, sometimes giving less 
than one-third the actual collapsing pressure. 

The general conclusions reached by Professor Stewart are thus 
stated by him: 

"1. The length of tube, between transverse joints tending to 
hold it to a circular form, has no practical influence upon the 
collapsing pressure of a commercial lap-welded steel tube, so 
long as this length is not less than about six diameters of tube. 

2. The formulas, as based upon the present research, for the 
collapsing pressure of modern lap-welded Bessemer steel tubes, 
are as follows: 



P== 1000(1 -^1-1600 ^2-) (A) 

P = 86,670^- 1386. (B) 

Where P = collapsing pressure, pounds per square inch 

d = outside diameter of tube in inches 

]{ = thickness of wall in inches. 
Formula (A) is for values of P less than 581 lb., or for values 

t 
of-^ less than 0.023, while formula (B) is for values greater than 

these. 

1 Trans. A. S. M. E., Vol. XXVII. 



STEEL TUBES 63 

These formulas, while strictly correct for tubes that are 20 ft. 
in length between transverse joints tending to hold them to a 
circular form, are, at the same time, substantially correct for 
all lengths greater than about six diameters. 

They have been tested for seven diameters, ranging from 
3 to 10 in., in all obtainable thicknesses of wall, and are known to 
be correct for this range. 

3. The apparent fiber stress under which the different tubes 
failed varied from about 7000 lb. for the relatively thinnest to 
35,000 lb. per square inch for the relatively thickest walls. 

Since the average yield-point of the material was 37,000 and 
the tensile strength 58,000 lb. per square inch, it would appear 
that the strength of a tube subjected to a collapsing fluid pressure 
is not dependent alone upon either the elastic limit or ultimate 
strength of the material constituting it.'' 

The following tables are condensed from those published by 
Professor Stewart and give average dimensions and pressures 
for each size tested, each result being the average of five tubes: 

The reader is referred to the published paper for further 
details of this most valuable contribution to a hitherto neglected 
subject. 

25. Theory. — In January, 1911, Professor Stewart presented a 
discussion of the theory of collapsed tubes based on the experi- 
ments above described.^ Considering a ring or annulus of the 
tube 1 in. long near the middle of its length, he treats each half of 
the ring as a column fixed at both ends and compressed uniformly 
along its center line, abc, Fig. 17. 

The ring is subjected to a uniform radial external pressure of 
p pounds per square inch and is therefore in the same condition 
as the thin shell in Art. 21 except that the resultant stress is now 
compression instead of tension. By equation (15), 



and 



^ pr pd 



2tS , , 

P = ^ (a) 



^ Trans. A. S. M. E., Vol. XXXIII. 



64 



MACHINE DESIGN 



TABLE XIX 

Collapsing Pressure of Tubes 





Average 


Average 


Actual 


Collapsing 


Test 


outside 


thickness of 


length of 


pressure, 


number 


diameter, 


wall. 


tube, 


pounds per 




inches 


inches 


feet 


square inch 


1 


8.643 


0.185 


20.026 


536 


2 


8.653 


0.184 


15.010 


548 


3 


8.656 


0.178 


10.002 


548 


4 


8.658 


0.180 


5.006 


592 


5 


8.656 


0.176 


2.512 


977 


6 


8.642 


0.215 


13.140 


847 


7 


8.663 


0.219 


11.801 


835 


8" 


8.669 


0.214 


10.007 


845 


9 


8.661 


0.212 


4.997 


907 


10 


8.657 


0.212 


2.507 


1,314 


11 


8.666 


0.267 


19.995 


1,438 


12 


8.652 


0.272 


14.996 


1,540 


13 


8.668 


0.267 


9.993 


1,533 


14 


8.656 


0.268 


4.993 


1,636 


15 


8.662 


0.268 


2.494 


1,784 


16 


8.657 


0.273 


19.387 


1,347 


17 


8.659 


0.275 


14.995 


1,421 


18 


8.671 


0.271 


10.003 


1,541 


19 


8.672 


0.280 


4.997 


1,731 


20 


8.653 


0.269 


2.505 


1,961 


21 


8.656 


0.294 


19.999 


1,686 


22 


8.654 


0.308 


14.987 


1,791 


23 


8.649 


0.305 


9.989 


1,810 


24 


8.654 


0.306 


4.993 


2,073 


25 


8.646 


0.311 


2.509 


2,397 


26 


6.017 


0.128 


20.000 


519 


27 


6.017 


0.131 


20.000 


529 


28 


6.022 


0.167 


20.000 


969 


29 


6.026 


0.166 


20.000 


924 


30 


6.032 


0.163 


20.000 


917 


31 


6.033 


0.170 


20 . 000 


1,007 


32 


6.023 


0.189 


20 . 000 


1,318 


33 


6.021 


0.212 


20.000 


1,457 


34 


6.015 


0.206 


20 . 000 


1,555 


35 


6.022 


0.186 


20.000 


1,188 


36 


6.032 


0.263 


20.000 


2,139 



STEEL TUBES 



65 



TABLE XIX— {Continued) 
Collapsing Pressure of Tubes 





Average 


Average 


Actual 


Collapsing 


Test 


outside 


thickness 


length of 


pressure, 


number 


diameter, 


of wall, 


tube, 


pounds per 




inches 


inches 


feet 


square inch 


37 


6.034 


0.264 


20.000 


2,381 


38 


6.654 


0.164 


20.000 


678 


39 


6.684 


0.200 


20.000 


1,184 


40 


6.666 


0.253 


20 . 000 


2,081 


41 


7.044 


0.160 


20.000 


563 


42 


7.050 


0.242 


20 . 000 


1,680 


43 


6.661 


0.154 


20.000 


563 


44 


6.655 


0.269 


20.100 


2,214 


45 


6.681 


0.249 


20.100 


1,745 


46 


6.049 


0.266 


20.110 


2,528 


47 


8.643 


0.185 


20.000 


536 


48 


8.642 


0.215 


14.133 


847 


49 


8.666 


0.267 


19.995 


1,438 


50 


8.657 


0.273 


19.550 


1,347 


51 


8.656 


0.293 


20 . 000 


1,686 


52 


8.663 


0.305 


20.100 


1,756 


53 


8.673 


0.354 


20.080 


2,028 


54 


6.987 


0.279 


20.170 


2,147 


55 


7.011 


0.160 


20 . 170 


621 


56 


5.993 


0.271 


20.180 


2,487 


57 


10.041 


0.165 


20.180 


225 


58 


10.026 


0.194 


20.110 


383 


59 


10.001 


0.316 


20.180 


1,319 


60 


3.993 


0.119 


20.170 


964 


61 


4.014 


0.175 


20.190 


2,280 


62 


4.026 


0.212 


20.190 


3,170 


63 


4.014 


0.327 


20.100 


5,560 


64 


3.000 


0.109 


20.000 


1,733 


65 


2.994 


0.113 


20.000 


1,962 


66 


2.992 


0.143 


20.000 


2,963 


67 


2.995 


0.188 


20.100 


4,095 


68 


10.779 


0.512 


19.470 


2,585 


69 


12.790 


0.511 


19.960 


2,196 


70 


13.036 


0.244 


20.000 


463 



66 



MACHINE DESIGN 



This stress is uniform from end to end as is the case with the 
loaded straight column in Fig. 18. Furthermore, the character- 
istic shape assumed by the collapsed tube, as shown in dotted 
lines in Fig. 17, has its tangents at a' and c' parallel to their 
original position at a and c, corresponding to the conditions for 
buckling of a column with fixed ends shown by dotted lines in 
Fig. 18. 




M 



\ 
\ 
\ 
\ 
\ 
\ \ 
\ \ 
\ \ 
\ \ 
\ \ 
\ I 

; 

/ / 
/ / 
/ / 



Fig. 17. 



Fig. 18. 



Let Z= length of equivalent column 

r = radius of gyration of section of column 

Then will l = %{d-t) 

(where d = outer diameter of tube) 
and 

[¥ ^ t 

^~\12~3.464 
By Professor Stewart's formula (B) 

P = 86,670-^- 1386 
a 

From (a) and (B) by equating: 

^=86,670:3-1386 
d ' d 



(b) 



(c) 



STEEL TUBES 67 

and 



ASf = 43,335 -693^ (d) 



From (b): 

d 21 , 

Substituting value of t from (c) : 

-,= oaL +1-0.1838 - + 1 (e) 

t 3.4647rr r 

Substituting this value of - in (d) and reducing: 

ASf = 42,642- 127.4- (24) 

corresponding to the straight line formula for columns (see 
Table la). 

Professor Stewart suggests as a substitute for formula (A) 
p. 62, the following: 

P = 50,210,000 (^y (G) 



26. Tube Joints. — The failure of boiler tubes, especially of those 
having water or steam pressure inside, is frequently due to 
slipping of the tube in the plate or fitting to which it joins. Such 
tubes are expanded in the plate by the use of a roller or Dudgeon 
expande'r and are sometimes flared or beaded on the outside for 
additional security. Under pressure, the tubes often slip in the 
holes so as to cause failure of the joint or at least leakage of the 
contained fluid. 

Some experiments made by Professors O. P. Hood and G. L. 
Christiansen were reported by them in 1908 and give the most 
reliable information on this subject.^ 

The tests were made on 3-in., twelve-gage, cold drawn Shelby 
tubes rolled into holes in plates of various thicknesses and reamed 
in various shapes. Some of the tubes were flared outside the 
plate and some not. 

Initial slip occurred at total pressures of from 5000 to 10,000 
lb. or from one-sixth to one-third the elastic limit of the material 

1 Trans. A. S. M. E., Vol. XXX. 



68 MACHINE DESIGN 

of the tube. The ultimate holding power was usually about 
double the slipping load. 

The coefficient of friction varied from 26 to 35 per cent, assum- 
ing the elastic limit to vary between 30,000 and 40,000 lb. 

The total friction per square inch of bearing area was about 
750 lb. Various degrees of rolling and various forms of tapered 
hole did not seem to affect the initial slipping load materially. 
Serrating the bearing surface of the hole had a very marked 
effect, raising the initial slipping load in some instances as high 
as 40,000 to 45,000 lb., or more than the elastic limit of the tube. 

The slipping point of the tube bears a certain analogy to the 
yield-point in metals and the diagrams of pressure and slip much 
resemble the stress-strain diagrams of soft steel. 

It is apparent from these experiments that overrolling has no 
advantages and that flaring the tubes will not prevent leakage. 

The fact that ordinarily slipping will occur at a pressure well 
inside the elastic limit of the material shows that timely warning 
will be given by leakage before there is any danger of failure. 

27. Tubes under Concentrated Loads. — In 1893, the author 
made some experiments on steel hoops to determine the strength 
and stiffness under a concentrated load applied in the direction 
of a diameter.^ 

Large steel tubes with relatively thin walls are sometimes 
exposed to external compression at the point of support causing 
distortion and occasionally permanent injury. 

The hoops tested were made of mild steel boiler plate, having 
a tensile strength of 60,000 lb. and a modulus of elasticity of 
30,000,000, cut into strips 2.5 in. wide, bent to a circular form 
and welded. Each hoop was compressed laterally in a testing 
machine until failure occurred, vertical and horizontal diameters 
being measured at regular intervals. 

Regarding the hoop as composed of two semi-circular columns 
fixed at the ends and each having a constant deflection of one- 
half the mean diameter, it is evident that a treatment is allowable 
similar to that used in Rankine's formula for columns (for- 
mula (12)). 

The increase in deflection for loads inside the elastic limit is 
small compared with the length of the hoop radius. 

^Jour. Assoc. Eng. Soc, Dec, 1893. 



STEEL HOOPS 69 

Let P = load in pounds at elastic limit 

D = inner horizontal diameter in inches 
b = breadth of hoop in inches 
t= thickness of ring 

aS= stress on inner fibers at extremity of horizontal 
diameter. 
Then as in Rankine's formula: 

^~2brbt'' ^^^ 

Where M is the bending moment at extremity of horizontal 
diameter. 

Assume M = kPD. 
Then 

where 5 = empirical constant. 

The average value of q as determined by experiment was 
g-0.946. 
Substituting this value in (b) and solving for P, we have: 

^" 1 + 0.946^' ^^^^ 

t 

Table XX gives the principal data and results of experiment. 

In determining the value of q from the experiments, S was 
assumed to be the same as the elastic limit in compression of a 
straight specimen of the same metal. 

The limited number of hoops tested and the method of their 
construction forbids the application of formula (25) to general 
cases of this character. It is offered here merely as a guide in 
design. 

28. Pipe Fittings. — Steam pipe up to and including pipe 2 in. 
in diameter is usually equipped with screwed fittings, including 
ells, tees, couplings, valves, etc. 

Pipe of a larger size, if used for high pressures, should be put 
together with flanged fittings and bolts. One great advantage of 



70 



MACHINE DESIGN 



m 

Pi 
o 






o 

02 



Average 
change 


d 


O 


1— ( 

d 


d 


CO 

o 


GO 

o 


O 

d 


Change 

inZ> 
under P 


d 


CO 

d 


o 
d 


I— 1 

d 


1> 

CO 

d 


00 
CO 

o 


o 


Change 

in D' 

under P 


d 


CO 

o 


O 


00 
I— 1 

d 


CO 

d 


00 

00 

d 


o 


Elastic 

limit 

of hoop P 


o 
o 


o 
o 
00^ 


O 

o 

CO 


o 
o 

co" 


o 
o 


o 
o 
■^^ 
i-T 


o 
o 
00 


Elastic 
limit 
compres- 
sion S 


o 

T— 1 
1—1 

CO 


O 

00^ 
rH 
CO 


o 

- 1—1 
00_ 

tH 

CO 


o 

1— 1 
00^ 
r-T 
CO 


O 
CO 


o 


o 

00 
CO 


Elastic 

limit 
tension 


o 
00 

CO 


O 
00 

d" 

CO 


o 

00 
cS 

CO 


o 

00 

<:S 

CO 


o 
o 

uo 


o 

00, 
CO 


o 
co^ 

CO" 
CO 


Thickness 
d 


Oi 
CO 

d 


Oi 
CO 

o 


o 
d 


CO 

d 


(M 

d 


o 

CO 

d 


d 


03 


0^ 

CO 


CO 


CO 
o 


CO 


CO 


CO 


CO 


Horizontal 

inside 

diameter 

D 


00 
CO 

I> 

1-1 


CSi 

1—1 


(M 

1— 1 
1-1 


05 
rH 

00 


o 

iH 


00 
1— 1 


o 

1— 1 


Vertical 

inside 

diameter 

D' 




CO 


I— 1 
1—1 


00 

o 
00 


CO 

o 
I— t 


o 

I— ( 

rH 


CO 

o 


6 


tH 


(M 


CO 


T^ 


lO 


CO 


I> 



PIPE FITTINGS 71 

the latter system is the fact that a section of pipe can easily be 
removed for repairs or alterations. 

Small connections are usually made of cast iron or malleable 
iron. While the latter are neater in appearance they are more 
apt to stretch and cause leaky joints. The larger fittings are 
made of cast iron or cast steel. Such fittings can be obtained in 
various weights and thicknesses, to correspond to those grades of 
pipe listed in the tables. 

The designer should have at hand catalogues of pipe fittings 
from the various manufacturers, as these will give in detail the 
proportions of all the different connections. 

For pressures not exceeding 100 lb. per square inch rubber and 
asbestos gaskets can be used between the flanges, but for higher 
pressures or for superheated steam, corrugated metallic gaskets 
are necessary. 

In 1905 some very interesting experiments on the strength of 
standard screwed elbows and tees were made by Mr. S. M. 
Chandler, a graduate of the Case School, and published by him 
in Power for October, 1905. 

The fittings were taken at random from the stock of the Pitts- 
burg Valve and Fittings Co., and three of each size were tested 
to destruction by hydraulic pressure. 

The following table gives a summary of the results obtained. 
The values which are starred in the table were obtained from 
fittings which had purposely been cast with the core out of center 
so as to make one wall thinner than the other. These values are 
not included in the averages. 

These tests show a large apparent factor of safety for any 
pressures to which screwed fittings are usually subjected. 

The failure of such fittings in practice must be attributed to 
faulty workmanship in erection, such as screwing too tight, lack 
of allowance for expansion and poor drainage. 

The average tensile strength of the cast iron used in the above 
fittings was 20,000 lb. per square inch. 

29. Flanged Fittings. — In 1907, the Crane Company published 
the results of a series of tests made on flanged tees and ells 
manufactured by that company.^ 

The fittings were tested by hydraulic pressure, a blank flange 

1 Valve World, Nov., 1907. 



72 



MACHINE DESIGN 



TABLE XXI 

Bursting Strength of Standard Screwed Fittings, Pressures in 

Pounds per Square Inch 



Size 




Elbows 




Average 


2i 


3,500 


3,300 


3,400 


3,400 


3 


2,400 


2,600 


2,100* 


2,500 


H 


2,100 


1,700* 


2,400 


2,250 


4 


2,800 


2,500 


2,500 


2,600 


4i 


2,000* 


2,600 


2,600 


2,600 


5 


2,600 


2,500 


2,500 


2,533 


6 


2,600 


2,200 


2,300 


2,367 


7 


1,800 


2,100 


1,900* 


1,950 


8 


1,700 


1,600 


1,700 


1,667 


9 


1,800 


1,800 


1,900 


1,833 


10 


1,800 


1,700 


1,600 


1,700 


12 


1,100 


1,200 


900* 


1,150 


Size 




Tees 




Average 


U 


3,400 


3,300 


3,300 


3,333 


U 


3,400 


3,200 


2,800* 


3,300 


2 


2,500 


2,800 


2,500 


2,600 


24 


2,400 


2,100* 


2,500 


2,450 


3 


1,400* 


1,900 


1,800 


1,850 


34 


1,200* 


1,500 


1,800 


1,650 


4 


1,800 


2,100 


1,700 


1,867 


44 


1,100* 


1,400 


1,400 


1,400 


5 


1,700 


1,300* 


1,500 


1,600 


6 


1,400 


1,500 


1,100* 


1,450 


7 


1,400 


1,400 


1,500 


1,433 


8 


1,200* 


1,400 


1,300 


1,350 


9 


1,300 


1,400 


1,200 


1,300 


10 


1,100 


1,300 


1,200 


1,200 


12 


1,100 


1,000 


1,100 


1,067 



* Made with eccentric core. 



PIPE FITTINGS 



73 







1-1 

6 



w 

Q 

Eh 
O 












Pi 

B 
^^ 
Q 

o 
o 



o 
6 



74 



MACHINE DESIGN 



being used to close the opening. Two materials were tried, 
cast iron having an average tensile strength of 22,000 lb. per 
square inch and ferro steel having a strength 50 per cent greater. 
The results are given as follows : 



TABLE XXII 

Strength of Flanged Fittings 
EXTRA HEAVY FITTINGS— TEES 



Size 
inches 



Body 
metal 
inches 



Burst ferro- 
steel lb. 

per 
sq. in. 



Average 



Burst cast 
iron lb. 

per 
sq. in. 



Average 



6 

6 

6 

8 

8 

8 

8 

8 

8 

10 

10 

10 

10 

10 

12 

12 

12 

12 

12 

12 

14 

14 

16 

16 

18 

18 

20 

20 

24 



2. 
4 

2. 

4 

2. 

4 

13 

15 

"16 

1.3 
1 6 

13 

16 

1 3 
16 

1 3 
16 

IS. 
1 6 

■II- 
1 6 

15 
16 

15 
1 6" 

15 
1 6 



1 

1 

1 

1 

1 

li 

li 

1t\ 

H 

li 

lA 

1^6 



2,700 
2,500 
3,000 
2,100 
2,250 
2,250 
2,100 
2,500 
2,300 
2,200 
2,200 
2,100 
2,000 
2,300 
2,200 
2,100 
2,000 
2,000 
2,100 
1,800 
1,900 
1,750 
1,700 
1,700 
1,600 
1,300 
1,400 
1,150 
1,300 



2,733 



2,250 



2,160 



2,033 
1,825 
1,700 



1,450 



1,275 
1,300 



1,675 
1,700 



1,200 
1,500 



1,225 
1,300 
1,200 
1,500 



1,100 
1,400 
1,500 
1,450 
1,450 



1,100 
1,050 
1,000 



600 



750 
700 



1,687 



1,350 



1,306 



1,380 
1,100 



1,025 
600 



750 
700 



PIPE FITTINGS 



75 



TABLE XXll— (Continued) 
EXTRA HEAVY FITTINGS— ELLS 



Size 
inches 


Body 
metal 
inches 


Burst ferro- 

steel lb. 

per 

sq. in. 


Average 


Burst cast 
iron lb. 

per 
sq. in. 


Average 


6 

6 

6 

8 

8 

8 

8 

10 

10 

10 

12 

12 

12 

14 

14 

16 


f 
f 

i 

ft 
it 
if 
If 
if 

15 

H 

n 
n- 


2,800 
3,500 
3,500 
2,700 
2,800 
2,800 
2,600 
2,550 
2,000 
2,500 
2,000 
2,200 
2,200 
1,700 








3,266 

2,725 
2,350 
2,133 


2,350 
2,200 
1,700 
1,600 
1,500 
1,700 
1,625 
1,400 
1,600 
1,275 
*900 
*700 
900 
1,250 
1,250 




2,275 






1,625 




1,541 




1,275 

1,075 
1,250 


2,100 





* Defective, eliminated from total. 

Strength of Flanged Fittings 
STANDARD CAST-IRON FITTINGS— TEES 



Size 
inches 


Body metal 
inches 


Bursting cast iron 
lb. per sq. in. 


Average 


6 
6 
8 
10 
12 
12 
14 
16 


f 

f 
i| 

if 
1 

1 


1,700 

1,500 

1,150 

1,100 

700 

850 

700 

750 




1,600 






775 






STANDARD CAST-IRON FITTINGS— ELLS 


6 
8 
10 
12 
14 
16 


f 

f 

if 

1 

1 


2,000 

1,500 

1,200 

900 

900 

850 

















76 



MACHINE DESIGN 



The Company recommends a rule which may be thus stated: 

P = ^ (26) 

where 

p = bursting pressure in lb. per square inch 
/S = tensile strength of metal 
t = thickness of wall in inches 
cl = inside diameter in inches 

c=a constant, 0.65 for sizes up to 12 in. and 0.60 for sizes 
above that. 




Fig. 21. 

A factor of safety of from 4 to 8 is recommended. 
The fractures were of various shapes and locations. The 
usual failure of the tees was by splitting in the plane of the axes 
from one flange to the next adjacent, Fig. 21. 

About half of the ells failed around a 
circumference inside one flange (Fig. 22) 
while six failed by splitting on the in- 
side of the bend. 

The effect on cast-iron fittings of 
high temperatures such as may occur 
with the use of superheated steam is not 
clearly understood. Professor Hollis 
and others report experiments on such 
fittings which seem to show some deteri- 
oration from this cause. ^ 
It is probable that most of the failures of pipe fittings in service 
are due to the excessive expansion and contraction of the pipe 
lines, incident to the use of high temperatures, rather than to the 
direct effect of the temperature or pressure, 
1 Trans. A. S. M. E., Vol. XXXI. 




Fig. 22. 



CYLINDERS 77 

A uniform temperature of 600 to 700° fahr. will not injure the 
cast-iron material, but where the temperature varies considerably, 
it is best to use some other metal. 

PROBLEMS 

1. Determine the bursting pressure of a wrought -iron steam pipe 6 in. 
nominal diameter, 

(a) If of standard dimensions. 

(6) If extra strong. 

(c) If double extra strong. 

2. Compare the above with the strength of standard screwed and standard 
flanged elbows and tees of the same size. 

3. Determine the probable collapsing pressure of a soft steel boiler-tube 
of 2 in. nominal diameter. 

4. Ditto, if tube is 6 in. in diameter. 

30. Steam Cylinders. — Cylinders of steam engines can hardly 
be considered as coming under either of the preceding heads. 
On the one hand the thickness of metal is not enough to insure 
rigidity as in hydraulic cylinders, and on the other the nature of 
the metal used, cast iron, is not such as to warrant the assump- 
tion of flexibility, as in a thin shell. Most of the formulas used 
for this class of cylinder are empirical and founded on modern 
practice. 

Van Buren's formula^ for steam cylinders is: 

t = .0001pd-\-.15Vd (27) 

A formula which the writer has developed is somewhat similar 
to Van Buren's. 

Let s'= tangential stress due to internal pressure. 
Then by equation for thin shells 

Let s'^ be an additional tensile stress due to distortion of the 
circular section at any weak point. 

Then if we regard one-half of the circular section as a beam 
fixed at A and B (Fig. 23) and assume the maximum bending 
moment as at C some weak point, the tensile stress on the outer 

^ See Whitham's ''Steam Engine Design," p. 27. 



78 



MACHINE DESIGN 



fibers at C due to the bending will be proportional to ^^ by 



the laws of flexure, or 



cpd^ 



where c is some unknown constant. 

The total tensile stress at C will then be 




S = s'+s''=^-{ 



pd cpd^ 



2t 



f 



Solving for c 
Solving for t 



t 



pd^ 
pd 



t 
~2d 



+ 



cpd"^ p^d^ 



(a) 

(28) 



a form which reduces to that of equation (15) when c = 0. . 

An examination of several engine cylinders of standard 
manufacture shows values of c ranging from .03 to .10, with an 
average value: 

c = M. 
The formula proposed by Professor Barr, in his paper on 
'' Current Practice in Engine Proportions,"^ as representing the 
average practice among builders of low-speed engines is: 

^ = .05d^-.3 in. (29) 

In Kent's Mechanical Engineer's Pocket Book, the following 
formula is given as representing closely existing practice: 

^ = .0004dp + 0.3 in. (30) 

» Trans. A. S. M. E., Vol. XVIII, p. 741. 



CYLINDERS 



79 



This corresponds to Barr's formula if we take p = 125 lb. per 
square inch. 

Experiments^ made at the Case School of Applied Science in 
1896-97 throw some light on this subject. Cast-iron cylinders 
similar to those used on engines were tested to failure by water 
pressure. The cylinders varied in diameter from 6 to 12 in. and 
in thickness from ^ to f in. 

Contrary to expectations most of the cylinders failed by tearing 
around. a circumference just inside the flange (see Fig. 24). 

Table XXIII gives a summary of the results. 



TABLE XXIII 



No, 


Diam. 
d 


Pres- 
sure 
P 


Thick- 
ness 
t 


Line of 
failure 


For 

15 

<; pd 


mulas use 
16 

r, pd 


d 

a 
c = 


Strength of 
test bar 


a 
d 
e 
f 
1 
2 
3 
4 
5 


12.16 

12.45 

9.12 

6.12 

9.58 

9.375 

9.13 

12.53 

12.56 


800 

700 

1,325 

2,500 

600 

1,050 

975 

700 

875 


.70 

.56 

.61 

.65 

.402 

.573 

.596 

.571 

.531 


Circum.. . . 

Longi 

Circum.. . . 
Circum.. . 
Longi. . . . 
Circum. . . 
Circum.. . 
Longi. . . . 
Circum.. . 


6,940 
7,780 
9,900 

11,800 
7,150 
8,590 
7,470 
7,680 

10,350 


3,470 

4,950 
5,900 

4,300 
3,740 

5,180 


.046 
.047 
.048 
.055 
.049 
.055 
.072 
.048 
.028 


18,000 lb. 
24,000 lb. 
24,000 lb. 
24,000 lb. 
24,000 lb. 
24,000 lb. 
24,000 lb. 
24,000 lb. 
24,000 lb. 



Average of c = .05 

Out of nine cylinders so tested, only three failed by splitting 
longitudinally. 

This appears to be due to two causes. In the first place, the 
flanges caused a bending moment at the junction with the shell 
due to the pull of the bolts. In the second place, the fact that 
the flanges were thicker than the shell caused a zone of weakness 
near the flange due to shrinkage in cooling, and the presence of 
what founders call "a hot spot.'' 

The stresses flgured from formula (16) in the cases where the 
failure was on a circumference, are from one-fifth to one-sixth 
the tensile strength of the test bar. 

1 Trans. A. S. M. E., Vol. XIX. 



80 



MACHINE DESIGN 




Fig. 24. — Fractured Cylinder. 




Fig. 25. — Fractured Cylinder. 



CYLINDERS 81 

The strength of a chain is the strength of the weakest link, and 
when the tensile stress exceeded the strength of the metal near 
some blow hole or ^^hot spot/' tearing began there and gradually 
extended around the circumference. 

Values of c as given by equation (a) have been calculated for 
each cylinder, and agree fairly well, the average value being 
c = .05. 

To the criticism that most of the cylinders did not fail by 
splitting, and that therefore formulas (a) and (22) are not appli- 
cable, the answer would be that the chances of failure in the two 
directions seem about equal, and consequently we may regard 
each cylinder as about to fail by splitting under the final pressure. 

If we substitute the average value of c = .05 and a safe value of 
/S = 2000, formula (28) reduces to: 



^^^ d j^^ (31) 

8000 ^200\^^ 1600 

An application of the Crane formula for cast-iron pipe fittings 
to some of the results in Table XXIII shows that the conditions 
are similar. 

6 St 
Using the formula p = '—r- for cylinders (d) and (4) in the table, 

we have approximately: 

_ .6X24,000X.57 _^^^ 

as against an actual value of 700. 

In a similar manner, testing cylinder (1) in table, we have: 

.65 X 24,000 X. 402 ^_ 

^= 9.58 = ^^^ 

as against 600 in the table. It will be noted that these are the 
cylinders which failed by splitting. 

Subsequent experiments^ made at the Case School in 1904 
show the effect of stiffening the flanges by brackets. 

Four cylinders were tested, each being 10 in. internal 
diameter by 20 in. long and having a thickness of about | in. 
The flanges were of the same thickness as the shell and were 
reenforced by sixteen triangular brackets as shown in Fig. 25. 

The fractures were all longitudinal there being but little of the 

1 Mchy., N. Y., Nov., 1905. 



82 



MACHINE DESIGN 



tearing around the shell which was so marked a feature of the 
former experiments. This shows that the brackets served their 
purpose. 

Table XXIV gives the results of the tests and the calculated 
values of c. 

TABLE XXIV 

Bursting Pressure of Cast-iron Cylinders 



Internal 


Average 


Bursting 


Value 


^_Vd 


diameter 


thickness 


pressure 


of c 


^ 2t 


10.125 


. 766 


1,350 


.0213 


9,040 


10.125 


0.740 


1,400 


.0152 


10,200 


10.125 


0.721 


1,350 


.0126 


9,735 


10.125 


0.720 


1,200 


.0177 


9,080 



Average value of c = .0167. 

Comparing the values in the above table with those in Table 
XXIII we find c to be only one-third as large. 

The tensile strength of the metal in the last four cylinders, as 
determined from test bars, was only 14,000 lb. per square inch. 

Comparison with the values of S due to direct tension as given 
by the formula 

pd 



S = 



2t 



shows that in a cylinder of this type about one-third of the stress 
is '^ accidental" and due to lack of uniformity in the conditions. 
In Table XXIII about two-thirds must be thus accounted for. 

PROBLEMS 

1. Referring to Table XXIII, verify in at least three experiments the 
values of S and c as there given. Do the same in Table XXIV. 

2. The steam cylinder of a Baldwin locomotive is 22 in. in diameter 
and 1.25 in. thick. Assuming 125 lb. gage pressure, find the value of c. 
Calculate thickness by Van Buren's and Barr's formulas. 

3. Determine proper thickness for cylinder of cast iron, if the diameter 
is 42 in. and the steam pressure 120 lb. by formulas 15, 27, 29, 30 and 31. 

4. The cylinder of a stationary engine has internal diameter = 14 in. and 
thickness of shell = 1.25 in. Find the value of c for p = 120 lb. per square 
inch. 



FLAT PLATES 83 

31. Thickness of Flat Plates. — An approximate formula for the 
thickness of flat cast-iron plates may be derived as follows: 
Let Z = length of plate in inches 
6 = breadth of plate in inches 
^ = thickness of plate in inches 
p = intensity of pressure in pounds 
iS = modulus of rupture pounds per square inch. 
A plate which is supported or fastened at all four edges is con- 
strained so as to bend in two directions at right angles. Now if 
we suppose the plate to be represented by a piece of basket work 
with strips crossing each other at right angles we may consider 
one set of strips as resisting one species of bending and the other 
set as resisting the other bending. We may also consider each 
set of strips as carrying a fraction of the total load. The equa- 
tion of condition is that each pair of strips must have a common 
deflection at the crossing. 

Suppose the plate to be divided lengthwise into flat strips an 
inch wide I inches long, and suppose that a fraction p' of the whole 
pressure causes the bending of these strips. 

Regarding the strips as beams with fixed ends and uniformly 
loaded: 

6M_ QWl _fP 

and the thickness necessary to resist bending is: 

= ; /Z. 



In a similar manner, if we suppose the plate to be divided into 
transverse strips an inch wide and b inches long, and suppose the 
remainder of the pressure p—p' equals p'' to cause the bending 
in this direction, we shall have: 

But as all these strips form one and the same plate the ratio 
of p' to p'' must be such that the deflection at the center of the 
plate may be the same on either supposition. The general for- 
mula for deflection in this case is 

~SS4:EI 



84 MACHINE DESIGN 

and 1= y^ for each set of strips. Therefore the deflection 
is proportional to ^-r^ and 3 in the two cases. 

But p' -\-p" = p 

Solving in these equations for p' and p" 

p¥ 



V 



l' + ¥ 
,,_ pi' 

V 74 j^ 7^4 



l^ + h' 
Substituting l^hese values in (a) and (b) 



'-'"'-Wm- ^''^ 



t = hl\ ^ • (33) 

As l>b usually, equation (33) is the one to be used. If the 
plate is square I = h and 

If the plate is merely supported at the edges then formulas 
(32) and (33) become: 
For rectangular plate: 



For square plate: 



^-^4w+hi)' ^^^^ 



--2X1- (^«) 



A round plate may be treated as square, with side = diameter, 
without sensible error. 

The preceding formulas can only be regarded as approximate. 
Grashof has investigated this subject and developed rational 
formulas but his work is too long and complicated for introduc- 
tion here. His formulas for round plates are as follows: 



FLAT PLATES 85 



Round plates: 
Supported at edges: 



Fixed at edges: 



-WS- (^^^ 



where t and p are the same as before, d is the diameter in inches 
and S is the safe tensile strength of the material. 

Comparing these formulas with (34) and (36) for square plates, 
they are seen to be nearly identical if allowance is made for the 
difference in the value of S. 

Experiments made at the Case School of Applied Science in 
1896-97 on rectangular cast-iron plates with load concentrated 
at the center gave results as follows: Twelve rectangular plates 
planed on one side and each having an unsupported area of 10 by 
15 in. were broken by the application of a circular steel plunger 
1 in. in diameter at the geometrical center of each plate. The 
plates varied in thickness from | in. to IJ in. Numbers 1 to 6 
were merely supported at the edges, while the remaining six were 
clamped rigidly at regular intervals around the edge. 

To determine the value of S, the modulus of rupture of the 
mateiial, pieces were cut from the edge of the plates and tested 
by cross-breaking. The average value of S from seven experi- 
ments was found to be 33,000 lb. per square inch. 

In Table XXV are given the values obtained for the breaking 
load W under the different conditions. 

If we assume an empirical formula: 

W = kj^, (a) 

and substitute given values of S, I and h we have nearly: 

W = lOOkt\ (b) 

Substituting values of W and t from the Table XXI we have 
the values of k as given in the last column. 

If we average the values for the two classes of plates and 
substitute in (a) we get the following empirical formulas: 



86 



MACHINE DESIGN 



For breaking load on plates supported at the edges and loaded 
at the center: 



and for similar plates with edges fixed: 



W = U2 



P + b' 



(39) 



(40) 



S in both formulas is the modulus of rupture. 

TABLE XXV 

Cast-iron Plates 10X15 in. 



No. 


Thickness t 


Breaking load W 


■ 
Constant k 


1 


.562 


7,500 


237 


2 


.641 


11,840 


288 


3 


.745 


14,800 


267 


4 


.828 


21,900 


320 


5 


1.040 


31,200 


289 


6 


1.120 


31,800 


254 


7 


.481 


9,800 


424 


8 


.646 


17,650 


422 


9 


.769 


26,400 


446 


10 


.881 


33,400 


430 


11 


1.020 


47,200 


454 


12 


1.123 


59,600 


477 



Those plates which were merely supported at the edges broke 
in three or four straight lines radiating from the center. Those 
fixed at the edges broke in four or five radial lines meeting an 
irregular oval inscribed in the rectangle. Number 12, however, 
failed by shearing, the circular plunger making a circular hole in 
the plate with several radial cracks. 

Some tests were made in the spring of 1906 at the Case School 
laboratories by Messrs. Hill and Nadig on the strength of flat 
cast-iron plates under uniform hydraulic pressure. 

Table XXVI gives the results of the investigation. 

The low value of S is explained by the fact that the material 
was a soft rather coarse gray iron, having an average tensile 
strength of about 12,000 lb. 



FLAT PLATES 



87 



TABLE XXVI 

Cast-iron Plates, Uniform Load, Fixed Edges 



Size of plate, 


Thickness 


Modulus 

S 


Breaking load in pounds per 
square inch 


inches 


Inches 


By formula 


Actual 


12X12 
12X12 
12X18 . 
12X18 


0.75 
1.00 
0.94 
1.25 


20,440 
27,900 
26,600 
24,000 


(34) 320 
(34) 777 
(33) 390 
(33) 622 


375 
675 
450 
650 



Further experiments are needed to establish any general 
conclusions. 



32. Steel Plates. — Mr. T. A. Bryson of Rensselaer Polytechnic 
Institute has recently made some tests on steel plates under 
hydrostatic pressure and published a monograph on the subject. 

The material tested was medium steel boiler plate from i to ^ 
in. thick and the sizes used were 18 by 18 in. and 24 by 24 in. 

Two plates separated by a cast-iron distance piece were 
clamped at the edges by cast-iron frames bolted together. 
Hydrostatic pressures from to 225 lb. per square inch were 
applied and deflections were measured at five points. Both 
working deflections and permanent sets wxre noted. The 
characteristics of the material were determined from test pieces 
cut off the edge of each plate. 

Mr. Bryson develops formulas similar to Morley's/ which 
differ from those just given in the values of the constants. All 
the formulas for square plates can, however, be reduced to the 
general form : 

t=b. ^ 



(See formula 34) 



or 



S^k 






(41) 



where S is the maximum stress in the plate. 

The value of k, as determined by the average of eight tests 
^ Morley's Strength of Materials, 



88 MACHINE DESIGN 

with different values of b and t, was 0.141 at the elastic limit of 
the material, the maximum value being 0.156 and the mini- 
mum 0.131. 

This value of k may then be used for steel plates with fixed 
edges without serious error. Mr. Bryson after discussing the 
experiments of Bach on square and rectangular plates recom- 
mends the following general formula for rectangular steel plates 
fixed at the edges and uniformly loaded : 

'^"l + 2.55r t' ^^"^^ 

where 

r = b/l 
Where l = h this reduces to formula (41). 

The value of S for plates merely supported may be assumed to 
be .50 per cent greater than in formula (42). 

The value of k in formula (32) is determined by substituting 

r = Y and reducing: 



i.e., k 



2(1 + r^) 



or for a square plate : 



S = l ^ (44) 



4 t 



These values of k are much larger than those just given. In 
Mr. Bryson's tests it was found that suspension stresses gradually 
supplanted those due to bending and that this change reduced 
the value of k. 

This would not be true of cast-iron plates and the formulas 
given on page 84 would be preferable. 

The values of k for the four experiments detailed in Table 
XXVI would be respectively: 

A; = .213-.287-.363-.400 
which shows that formula (42) is not applicable to cast-iron 
plates. 

The most comprehensive experiments on flat plates are those 
by Professor Bach, and Grashof's formulas are largely controlled 



FLAT PLATES 



89 



by them.^ Table XXVII gives the derived formulas for some 
of the more usual cases. The notation is the same as that of 
the previous formulas. 

The strength of the plates depends also on the manner of 
fastening at the edges^ the number and size of bolts, the nature 
of gasket used, if any, etc.,, etc. 

TABLE XXVII 
Stresses in Flat Plates 



Shape 


Edges 


Load 


Value of 

fiber stress 

S = 


Value of 

coefficient 

k = 


Remarks 


Circle 


Fixed 


Uniform. . . . 


1^ i2 


Cast iron, 0.8 

Steel, 0.5 


r = radius. 




Circle 


Support.. . 


Uniform. . . . 


pr2 


Cast iron, 1.2 

Steel, 0.7 


r = radius. 


Ellipse. . . . 


Fixed 


Uniform. . . . 




Cast iron, 1.34 

Steel, 0.84 ... 


Estimated. 




Ellipse 


Support. . . 


Uniform. . . . 




Cast iron, 2.26 

Steel, 1.41 


Estimated. 




Rect 


Fixed 


At center. . . 


Wlb 


Cast iron, 2.63 








Rect 


Support. . . 


At center. . . 


Wlb 

t^l^+b^) 


Cast iron, 3.0 








Rect 


Fixed 


Uniform. . . . 


pl^ 62 

tH P+ 62) 


Cast iron, 0.38 

Steel, 0.24 


Estimated. 




Rect 


Support. . . 


Uniform .... 


- pZ2 62 
^2(Z2+62) 


Cast iron, 0.57 

Steel 36 


Estimated. 




Square. . . . 


Fixed 


At center. . . 


^<2 


Cast iron, 1.32. . . . 








Square.. . . 


Support. . . 


At center. . . 


*? 


Cast iron 1 50 . . 








Square. . . . 


Fixed 


Uniform .... 


*f 


Cast iron, 0.19 

Steel, 0.12 


Estimated. 




Square.. . . 


Support. . . 


Uniform .... 


,p62 

^ t 


Cast iron, 0.28 

Steel, 0.18 


Estimated. 





minor axis 



Note. — n ■ 

major axis 

1 See Am. Mach., Nov. 25, 1909. 



90 MACHINE DESIGN 

It will be interesting to compare values of S in Table XXVII 
with those obtained by experiment so as to determine whether 
S corresponds to the tensile strength of the metal or to the 
modulus of rupture in cross breaking. 

PROBLEMS 

1. Calculate the thickness of a steam-chest cover 12X16 in. to sustain 
a pressure of 90 lb. per square inch with a factor of safety = 10. 

2. Calculate the thickness of a circular manhole cover of cast iron 18 in. 
in diameter to sustain a pressure of 200 lb. per square inch with a factor of 
safety = 8, regarding the edges as merely supported. 

3. Determine the probable breaking load for a plate 18X24 in. loaded 
at the center, (a) when edges are fixed. (6) When edges are supported. 

4. In experiments on steam cylinders, a head 12 in. in diameter and 1.18 
in. thick failed under a pressure of 900 lb. per square inch. Determine the 
value of S by formula (34). 

REFERENCES 

Mechanics of Materials, Merriman, Chapter XIV. 

Strength of Materials, Slocum and Hancock, Chapters VII and VIII. 

Details of High-pressure Piping. Cass. June, 1906. 

Design and Construction of Piping. Eng. Mag., April, 1908. 

Piping for High Pressures. Power, Sept. 22, 1908. 

Flanges for High Pressures. Power, July, 1905; Dec, 1905. 

High-pressure Tests of Large Pipes. Eng. News, Apr. 15, 1909. 



CHAPTER IV 
FASTENINGS 

33. Bolts and Nuts. — Tables of dimensions for U. S. standard 
bolt heads and nuts are to be found in most engineering hand- 
books and will not be repeated here. 

These proportions have not been generally adopted on account 
of the odd sizes of bar required. The standard screw-thread has 
been quite generally accepted as superior to the old V-thread. 

Roughly the diameter at root of thread is 0.83 of the outer 
diameter in this system^ and the pitch in inches is given by the 
formula 



p-.24\/^ + .625-.175 



(45) 



where d = outer diameter. 

TABLE XXVIII 

Safe Working Strength of Iron or Steel Bolts 



Diam. 

of bolt, 
inch 


Threads 
per inch. 

No. 


Diam. at 
root of 
thread, 
inches 


Area at 
root of 
thread, 
sq. in. 


Safe load in tension, 
pounds 


Safe load in shear, 
pounds 


5,000 lb. 
per sq. in. 


7,500 lb. 
per sq. in. 


4,000 lb. 
per sq. in. 


6,000 lb. 
per sq. in. 


1 


20 
18 
16 
14 
13 


.185 
.240 
.294 
.344 
.400 


.0269 
.0452 
.0679 
.0930 
.1257 


135 
226 
340 
465 
628 


202 
340 
510 
695 
940 


196 
306 
440 
600 

785 


294 
460 
660 
900 
1,175 



91 



92 



MACHINE DESIGN 



TABLE XXVIII {Continued) 
Safe Working Strength of Iron or Steel Bolts 











Safe load in tension, 


Safe load 


in shear. 


Diam. 

of bolt, 

inch 


Threads 

per inch, 

No. 


Diam at 
root of 
thread, 
inches 


Area at 
root of 
thread, 
sq. in. 


pounds 


pounds 


5,000 lb. 


7,500 lb. 


4,000 lb. 


6,000 lb. 










per sq. in. 


per sq. in. 


per sq. in. 


per sq. in. 


i%- 


12 


.454 


.162 


810 


1,210 


990 


1,485 


5 

8 


11 


.507 


.202 


1,010 


1,510 


1,2.30 


1,845 


f 


10 


.620 


.302 


1,510 


2,260 


1,770 


2,650 


7 
8 


9 


.731 


.420 


2,100 


3,150 


2,400 


3,600 




8 


.837 


.550 


2,750 


4,120 


3,140 


4,700 


u 


7 


.940 


.694 


3,470 


5,200 


3,990 


6,000 


u 


7 


1.065 


.891 


4,450 


6,680 


4,910 


7,360 


If 


6 


1.160 


1.057 


5,280 


7,920 


5,920 


7,880 


u 


6 


1.284 


1.295 


6,475 


9,710 


7,070 


10,600 


If 


5i 


1.389 


1.515 


7,575 


11,350 


8,250 


12,375 


If 


5 


1.490 


1.744 


8,720 


13,100 


9,630 


14,400 


1 '7 

-•^ 8 


5 


1.615 


2.049 


10,250 


15,400 


11,000 


16,500 


2 


4i 


1.712 


2.302 


11,510 


17,250 


12,550 


18,800 



The shearing load is calculated fiom the area of the body of 
the bolt. 

Bolts may be divided into three classes which are given in the 
order of their merit. 

1. Through bolts, having a head on one end and a nut on the 
other. 

2. Stud bolts, having a nut on one end and the other screwed 
into the casting. 

3. Tap bolts or screws having a head at one end and the other 
screwed into the casting. 

The principal objection to the last two forms and especially to 
(3) is the liability of sticking or breaking off in the casting. 

Any irregularity in the bearing sui faces of head or nut where 
they come in contact with the casting, causes a bending action 
and consequent danger of rupture. 

This is best avoided by having a slight annular projection on 
the casting concentric with the bolt hole and finishing the flat 
surface by planing or counter-boring. 

Counter-boring without the projection is a rather slovenly way 
of over coming the difficulty. 



BOLTS AND NUTS 



93 



a 



\}f7 



b 



When bolts or studs are subjected to severe stress and vibration, 
it is well to turn down the body of the bolt to the diameter at 
root of thread, as the whole bolt will then stretch slightly under 
the load. 

A check nut is a thin nut screwed firmly against the main nut 
to prevent its working loose, and is commonly placed outside. 

As the whole load is liable to come on the outer nut, it would 
be more correct to put the main nut outside. (Prove this by 
figure.) 

After both nuts aie firmly screwed down, the outer one should 
be held stationary and the inner one reversed against it with 
what force is deemed safe, that the greater reaction may be 
between the nuts. 

Numerous devices have been invented for the purpose of hold- 
ing nuts from working loose under vibration but none of them are 
entirely satisfactory. 

Probably the best method for large 
nuts is to drive a pin or cotter entirely 
through nut and bolt. 

A flat plate, cut out to embrace the 
nut and fastened to the casting by a 
machine screw, is often used. 

Machine Screws. — A screw is distin- 
guished from a bolt by having a slot- 
ted, round head instead of a square or 
hexagon head. 

The head may have any one of four Yig. 26. 

shapes, the round, fillister, oval fillister 

and flat as shown in Fig. 26. A committee of the American 
Society of Mechanical Engineeis has recently recommended 
certain standards for machine screws. The form of thread 
recommended is the U. S. Standard or Sellers type with provi- 
sion for clearance at top and bottom to insure bearing on the 
body of the thread. 

The sizes and pitches recommended are shown in Table XXIX. 

In designing eye-bolts it is customary to make the combined 
sectional area of the sides of the eye one and one-half-times that 
of the bolt to allow for obliquity and an uneven distribution of 
stress. 



ca 






94 



MACHINE DESIGN 



TABLE XXIX 
Machine Screws 



Standard diam. 


.070 


.085 


.100 


.110 


.125 


.140 


.165 


.190 


.215 


.240 


.250 


.270 


.320 


.375 


Threads per in. 


72 


64 


56 


48 


44 


40 


36 


32 


28 


24 


24 


22 


20 


16 



Reference is made to the report itself for further details of 
heads, taps, etc. 

34. Crane Hooks. — Heretofore, the large wrought-iron or steel 
hooks used for crane service have usually been designed by con- 
sidering the fibers on the inside of a hook to be subjected to a 
tension which was the resultant of the direct load and of the 
bending due to the eccentricity of the loading. 

Experiments made by Professor Rautenstrauch in 1909^ 
show that such methods do not give correct results. Ten hooks 
of various capacities were tested by direct loading and their 
elastic limits determined. 

The following table gives the leading data and results. The 
dimensions are those of the principal cross-section: 

TABLE XXX 
Elastic Limit of Crane Hooks 



Nominal 

capacity, 

tons 


Material 


Cross-section dimensions 


Elastic 

limit, 

lb. 


A 


/ 


I 


y 


30 

20 

15 

15 

10 

10 

5 

5 

3 

2 


C. steel . . . 
C. steel . . . 
C. steel . . . 
W. iron . . . 
C. steel. . . 
W. iron . . . 
C. steel . . . 
W. iron . . . 
C. steel . . . 
C. steel . . . 


23.35 
14.48 
13.92 
8.40 
8.72 
6.08 
5.69 
4.80 
3.50 
2.03 


111.6 

11.9 
6.5 

3.8 


7.25 
5.90 
5.13 
5.00 
4.30 
4.00 
3.25 
3.47 
2.89 
2.03 


3.36 
2.75 
2.23 
1.87 
2.05 
1.50 
1.42 
1.35 
1.16 
0.88 


56,000 
30,000 
48,000 
16,000 
18,000 
16,000 
18,000 
14,000 
8,500 
4,700 



» Am. Mach., Oct. 7, 1909. 



CRANE HOOKS 



95 



A = area in square inches 
/ = moment of inertia about gravity axis 
Z = distance from load line to gravity axis 
2/ = distance from inner fiber to gravity axis. 

It will be noticed that the nominal capacity of the hook is in 
several cases greater than the elastic limit as shown by experi- 
ment. This is particularly true ot the larger sizes. 

The standard cross-section of crane hooks is that of a trapezoid 
with curved bases as shown in Fig. 27. The wider base corre- 
sponds to the inner side of the hook where 
the tension is greatest. 

The dimensions given are approxi- 
mately those of a 20-ton steel hook. 
Professor Rautenstrauch finds that the 
values of the load at elastic limit, as de- 
termined by the ordinary formula above 
alluded to, are entirely erroneous, being 
in many cases more than twice that found 
by the actual tests. He recommends in- 
stead the so-called Andrews-Pearson for- 
mula which takes into account the curva- 
ture of the neutral axis and the lateral 
distortion of the metal. 

The discussion is too long for reproduc- 
tion here and reference is made to his paper 
and to the original presentation of this 
formula. ^ 

A similar condition exists in large chain links. The bending 
moment in this case is, however, usually eliminated by the 
insertion of a cross piece or strut. ^ 

PROBLEMS 

1. Discuss the effect of the initial tension caused by the screwing up of 
the nut on the bolt, in the case of steam fittings, etc.; i.e., should this tension 
be added to the tension due to the steam pressure, in determining the proper 
size of bolt? 

^ Technical Series 1, Draper Company's Research Memoirs, 1904. See 
also Slocum and Hancock's Strength of Materials. 

2 See Univ. of HHnois, Bulletin No. 18. " The Strength of Chain Links," 
by G. A. Goodenough and L. E. Moore. 




Fig. 



27.— 20-ton steel 
hook. 



96 



MACHINE DESIGN 



2. Determine the number of |-in. steel bolts necessary to hold on the 
head of a steam cylinder 18 in. diameter, with the internal pressure 90 lb. 
per square inch, and factor of safety = 12. 

3. Show what is the proper angle between the handle and the jaws of a 
fork wrench. 

(1) If used for square nuts. 

(2) If used for hexagon nuts; illustrate by figure. 

4. Determine the length of nut theoretically necessary to prevent stripping 
of the thread, in terms of the outer diameter of the bolt. 

(1) With U. S. standard thread. 

(2) With square thread of same depth. 

5. Design a hook with a swivel and eye at the top to hold a load of 10 
tons with a factor of safety 5, the center line of hook being 8 in. from line of 
load, and the material soft steel. 

35. Riveted Joints. — Riveted joints may be divided into two 
general classes: lap joints where the two plates lap over each 

other, and butt joints where 



the edges of the plates butt 
together and are joined by 
B over-lapping straps or welts. 
If the strap is on one side 
only, the joint is known as a 
butt joint with one strap: if 
straps are used inside and 
out the joint is called a butt 
joint with two straps. Butt 

JL 

2 





Fig. 28. 



joints are generally used when the material is more than 
in. thick. 

Any joint may have one, 
two or more rows of rivets 
and hence be known as a 
single riveted joint, a double 
riveted joint, etc. 

Any riveted joint is weaker 
than the original plate, simply 
because holes cannot be 
punched or drilled in the 
plate for the introduction of 
rivets without removing some of the metal. 

Fig. 28 shows a double riveted lap joint and Fig. 29 a single 
riveted butt joint with two straps. 




o 


o 


^ 


o 


o 


o < 










Fig. 29. 



RIVETED JOINTS 



97 



Riveted joints may fail in any one of four ways: 

1. By tearing of the plate along a line of rivet holes, as at AB, 
Fig. 28. 

2. By shearing of the rivets. 

3. By crushing and wrinkling of the plate in front of each rivet 
as at C, Fig. 28, thus causing leakage. 

4. By splitting of the plate opposite each rivet as at D, Fig. 28. 
The last manner of failure may be prevented by having a suffi- 
cient distance from the rivet to the edge of the plate. Practice 
has shown that this distance should be at least equal to the 
diameter of a rivet. 

Experience has shown that lap joints in plates of even moderate 
thickness are dangerous on account of the liability of hidden 
cracks. Several disastrous boiler explosions have resulted from 
the presence of cracks inside the joint which could not be detected 
by inspection. The fact that one or both plates are out of the 
line of pull brings a bending moment on both plates and rivets. 

Some boiler inspectors have gone so far as to condemn lap 
joints altogether. 

Let t = thickness of plate 

d = diameter of rivet hole 

p = pitch of rivets 

n= number of rows of rivets 

7" = tensile strength of plate 

C = crushing strength of plate or rivet 

/S=shearing strength of rivet. 

Average values of the constants are as follows: 



Material 


T 


c 


s 


Wrought iron 

Soft steel 


50,000 
56,000 


80,000 
90,000 


40,000 
45,000 





The values of the constants given above are only average 
values and are liable to be modified by the exact grade of material 
used and the manner in which it is used. 

The tensile strength of soft steel is reduced by punching from 



98 MACHINE DESIGN 

3 to 12 per cent according to the kind of punch' used and the 
width of pitch. The shearing strength of the rivets is diminished 
by their tendency to tip over or bend if they do not fill the holes, 
while the bearing or compression is doubtless relieved to some 
extent by the friction of the joint. The values given allow 
roughly for these modifications. 

36. Lap Joints. — This division also includes butt joints which 
have but one strap. ^ 

Let us consider the shell as divided into strips at right angles 
to the seam and each of a width = p. Then the forces acting on 
each strip are the same and we need to consider but one strip. 

The resistance to tearing across of the strip between rivet holes 

is (p-d)tT (a) 

and this is independent of the number of rows of rivets. 
The resistance to compression in front of rivets is 

ndtC (b) 

and the resistance to shearing of the rivets is 

"^nd'S. (c) 

If we call the tensile strength 7" = unity then the relative 
values of C and S are 1.6 and 0.8 respectively. 

Substituting these relative values of T, C and S in our equations, 
by equating (b) and (c) and reducing we have 

d = 2Mt (46) 

Equating (a) and (c) and reducing we have 

p = d + .Q28^ (47) 

Or by equating (a) and (b) 

p = d + lMd (48) 

These proportions will give a joint of equal strength throughout, 
for the values of constants assumed. 

37. Butt Joints with Two Straps. — In this case the resistance to 
shearing is increased by the fact that the rivets must be sheared 



RIVETED JOINTS 99 

at both ends before the joint will fail. Experiment has shown 
this increase of shearing strength to be about 85 per cent and we 
can therefore take the relative value of aS as 1.5 for butt joints. 
This gives the following values for d and p 

d = lMt (49) 

7)d^ 

p = d-{-l.lS^ (50) 

p = d + lMd. (51) 

In the preceding formulas the diameter of hole and rivet have 
been assumed to be the same. 

The diameter of the cold rivet before insertion will be y^g- in. 
less than the diameter given by the formulas. 

Experiments made in England by Prof. Kennedy give the 
following as the proportions of maximum strength: 

Lap joints d = 2.S3t 

p = d + 1.375nd 

Butt joints d = l.St 

p = d + 1.55nd 



38. Efficiency of Joints. — The efficiency of joints designed like 
the preceding is simply the ratio of the section of plate left 
between the rivets to the section of solid plate, or the ratio of the 
clear distance between two adjacent rivet holes to the pitch. 
From formula (48) we thus have : 

1 G??- 
Efficiency = ^-^^. (52) 

This gives the efficiency of single, double and triple riveted 
seams as 

.615, .762 and .828 respectively. 

Notice that the advantage of a double or triple riveted seam 
over the single is in the fact that the pitch bears a greater ratio 
to the diameter of a rivet, and therefore the proportion of metal 
removed is less. 



100 



MACHINE DESIGN 



39. Butt Joints with Unequal Straps. — One joint in common use 
requires special treatment. 

It is a double riveted butt joint in which the inner strap is 
made wider than the outer and an extra row of rivets added, 
whose pitch is double that of the original seam; this is sometimes 
called diamond riveting. See Fig. 30. 

This outer row of rivets is 



then exposed to single shear 
and the original rows to dou- 
ble shear. 

Consider a strip of plate of 
a width = 2p. Then the resis- 
tance to tearing along the 
outer row of rivets is 



C 



o o 


o o ol 


o o o / 


o o o 


O O O (j 


o o/ 


J==_=^ 



(2p-d)tT 

As there are five rivets to 
compress in this strip the 
bearing resistance is 

5dtC 



Fig. 30. 



As there is one rivet in 
single shear and four in double shear the resistance to shearing is 

|l + (4Xl.85) I 'yi'S = 6M'S 

Solving these equations as in previous cases, we have for this 
particular j oint 

d = 1.52t (53) 

2p = 9d 
p = 4:M (54) 

2 p-d _^ 
2p ~~ 9' 



Efficiency = 



(55) 



40. Practical Rules. — The formulas given above show the 
proportions of the usual forms of joints for uniform strength. 

In practice certain modifications are made for economic reasons. 
To avoid great variation in the sizes of rivets the latter are graded 
by sixteenths of an inch, making those for the thicker plates con- 



RIVETED JOINTS 



101 



siderably smaller than the formula would allow, and the pitch is 
then calculated to give equal tearing and shearing strength. 

Table XXXI shows what may be considered average practice 
in this country for lap joints with steel plates and rivets. 



TABLE XXXI 
Riveted Lap Joints 



Thick- 


Diam. 


Diam. 


Pitch 


Efficiency 


of plate 


ness of 


of 


of 
















plate 


rivet 


hole 


Single 


Double 


Single 


Double 


i 


i 


T^6 


If 


If 


.59 


.68 


5 
1 6 


f 


H 


If 


2i 


.58 


.68 


f 


f 


H 


1| 


2i 


.57 


.67 


7_ 
16 


13 
16 


1 


2 


2| 


.56 


.68 


h 


1 


n- 


2 


2| 


.53 


.67 



The efficiencies are calculated from the strength of plate 
between rivet holes and the efficiencies of the rivets may be even 
lower. Comparing these values with the ones given in Art. 38 
we find them low. This is due to the fact that the pitches 
assumed are too small. The only excuse for this is the possibility 
of getting a tighter joint. 

TABLE XXXII 
Riveted Butt Joints 











Pitch 




Thickness of 


Diam. of 


Diam. of 








plate 


rivet 


hole 














Single 


Double 


Triple 


i 


f 


13. 
I 6 


21 


4 


5i 


f 


13 
16 


1 


2f 


31 


5i 


1 


1 


1.5 
16 


2f 


31 


5i 


i 


JL5. 
16 


1 


2| 


3f 


5 


1 


1 


ItV 


2| 


31 


5 



102 



MACHINE DESIGN 



Table XXXII has been calculated for butt joints with two 
straps. As in the preceding table the values of the pitch are too 
small for the best efficiency. The tables are only intended to 
illustrate common practice and not to serve as standards. There 
is such a diversity of practice among manufacturers that it is 
advisable for the designer to proportion each joint according to 
his own judgment, using the rules of Arts. 36-39 and having 
regard to the practical considerations which have been mentioned. 

A committee of the Master Steam Boiler Makers' Association 
has made a number of tests on riveted joints and reported its 
conclusions. The specimens were prepared according to generally 
accepted practice, but on subjecting them to tension many of 
them failed by tearing through from hole to edge of plate. The 
committee recommends making this distance greater, so that 
from the center of hole to edge of plate shall be perhaps 2d instead 
of IM. 

The committee further found the shearing strength of rivets 
to be in pounds per square inch of section. 





Single shear 


Double shear 


Iron rivets 


40,000 
49,000 


78,000 
84,000 


Steel rivets 





Compare these values with those given in Art. 35. Also note 
that the factor for double shear is 1.95 for iron rivets and only 
1.71 for steel rivets as against the 1.85 given in Art. 37. The 
committee found that machine-driven rivets were stronger in 
double shear than hand-driven ones. 

PROBLEMS 

1. Calculate diameter and pitch of rivets for ^-in. and ^-in. plate and 
compare results with those in Table XXXI. Criticise latter. 

2. Show the effect in Prob. 1 of using iron rivets in steel plates. 

3. Criticise proportions of joints for ^-in. and 1-in. plate in Table XXXII 
by testing the efficiency of rivets and plates. 

4. A cylinder boiler 5X16 ft. is to have long seams double riveted laps 
and ring seams single riveted laps. If the internal pressure is 90 lb. gage 
pressure and the material soft steel, determine thickness of plate and pro- 
portion of joints. The net factor of safety at joints to be 5. 



RIVETED JOINTS 



103 



5. A marine boiler is 13 ft. 6 in. in diameter and 14 ft. long. The long 
seams are to be diamond riveted butt joints and the ring seams ordinary 
double riveted butt joints. The internal pressure is to be 175 lb. gage and 
the material is to be steel of 60,000 lb. tensile strength. Determine thick- 
ness of shell and proportions of joints. Net factor of safety to be 5, as in 
Prob. 4. 




Fig. 31. 

6. Design a diamond riveted joint such as shown in Fig. 31 for a steel 
plate f in. thick. Outer cover plate is f in. and inner cover plate is ^^ in. 
thick; the pitch of outer rows of rivets to be twice that of inner rows. 
Determine efficiency of joints. 

7. The single lap joint with cover plate, as shown in Fig. 32, is to have 
pitch of outer rivets double that of inner row. Determine diameter and 
pitch of rivets for |-in. plate and the efficiency of joint. 




Fig. 32. 



41. Riveted Joints for Narrow Plates. — The joints which have 
been so far described are continuous and but one strip of a width 
equal to the pitch or the least common multiple of several 
pitches, has been investigated. 

When narrow plates such as are used in structural work are 
to be joined by rivets, the joint is designed as a whole. Diamond 
riveting similar to that shown in Fig. 30 is generally used and the 
joint may be a lap, or a butt with double straps. The diameter of 
rivets may be taken about 1.5 times the thickness of plate [see 
equation (53)], and enough rivets used so that the total shearing 
strength may equal the tensile strength of the plate at the point 
of the diamond, where there is one rivet hole. It may be neces- 
sary to put in more rivets of a less diameter in order to make 
the figure symmetrical. p. 



104 



MACHINE DESIGN 



The efficiency of the joint may be tested at the different rows 
of rivets, allowing for tension of plate and shear of rivets in each 
case. 

9 

PROBLEMS 

1. Design a diamond riveted lap joint for a plate 12 in. wide and f in. 
thick, and calculate least efficiency for shear and tension. 
^ 2. A diamond riveted butt joint with two straps has rivets arranged as in 
Fig. 33, the plate being 12 in, wide and | in. thick, and the rivets being 1 
in. in diameter. If the plate and rivets are of steel, find the probable 
ultimate strength of the following parts : 
(a) The whole plate. 
(6) All the rivets on one side of the joint. 

(c) The joint at the point of the diamond. 

(d) The joint at the row of rivets next the point. 

42. Joint Pins. — A joint pin is a bolt exposed 
to double shear. If the pin is loose in its bear- 
ings it should be designed with allowance for 
bending, by adding from 30 to 50 per cent to the 
area of cross-section needed to resist shearing 
alone. Bending of the pin also tends to spread 
apart the bearings and this should be prevented 
by having a head and nut or cotter on the pin. 

If the pin is used to connect a knuckle joint 
as in boiler stays, the eyes forming the joint 
should have a net area 50 per cent in excess of 

the body of the stay, to allow for bending and uneven tension 

(see Eye-bolts, Art. 33). 
Fig. 34 shows a pin and angle joint for attaching the end of a 

boiler stay to the head of the boiler. 

43. Cotters. — A cotter is a key which passes diametrically 
through a hub and its rod or shaft, to fasten them together, and 
is so called to distinguish it from shafting keys which lie parallel 
to axis of shaft. 

Its taper should not be more than 4 degrees or about 1 in 15, 
unless it is secured by a screw or check nut. 

The rod is sometimes enlarged where it goes in the hub, so that 
the effective area of cross-section where the cotter goes through 
may be the same as in the body of the rod. (See Fig. 35.) 




Fig. 33. 



COTTERS 



105 



Let: d = diameter of body of rod 

d^ = diameter of enlarged portion 

f= thickness of cotter, usually = 



di 



b = breadth of cotter 

Z= length of rod beyond cotter. 

Suppose that the applied force is a pull on the rod — causing 
tension on the rod and shearing stress on the cotter. 
The effective area of cross-section of rod at cotter is 



t 




/I 



j^ 



^ 




w 



VJ 



CZZ] 




Fig. 34. 



Fig. 35. 



Ttd^' 



d^ d^ 

and this should equal the area of cross-section of the body of rod. 



^i=^^/^= 



1.21d. 



(56) 



Let P = pull on rod. 

*S^ = shearing strength of material. 
The area to resist shearing of cotter is 



h = 



2P 

d,S' 



(a) 



The area to resist shearing of rod is 

P 

S 



2d,l 



106 



MACHINE DESIGN 



and 1 = 



2d,S 



(b) 



If the metal of rod and cotter are the same 



2d,l = 



1 = 



b. 



(57) 



r 




II 



1 



■^ 



u 



y 



J 



Fig. 36. 



Great care should be taken in fitting cotters that they may 
not bear on corners of hole and thus tear the rod in two. 

A cotter or pin subjected to alternate 
stresses in opposite directions should 
have a factor of safety double that 
otherwise allowed. 

Adjustable cotters, used for tighten- 
ing joints of bearings are usually ac- 
companied by a gib having a taper equal 
and opposite to that of the cotter (Fig. 
36). In designing these for strength 
the two can be regarded as resisting 
shear together. 
For shafting keys see chapter on shafting. 
The split pin is in the nature of a cotter but is not usually 
expected to take any shearing stress. 

PROBLEMS 

1. Design an angle joint for a soft steel boiler stay, the pull on stay being 
12,000 lb. and the factor of safety, 6. Use two standard angles. 

2. Determine the diameter of a round cotter pin for equal strength of 
rod and pin. 

3. A rod of wrought iron has keyed to it a piston 24 in. in diameter, by a 
cotter of machinery steel. 

Required the two diameters of rod and dimensions of cotter to sustain a 
pressure of 150 lb. per square inch on the piston. Factor of safety = 8. 

4. Design a cotter and gib for connecting rod of engine mentioned in 
Prob. 3, both to be of machinery steel and .75 in. thick. (See Fig. 36.) 

REFERENCES 

Machine Design. Unwin., Vol. I, Chapters IV and V. 
Failures of Lap Joints. Power, May, 1905; Feb., 1907; Nov., 1907. 
Tests of Nickel Steel Riveted Joints. By A. N. Talbot and H. F. Moore. 
University of Illinois Bulletin, No. 49. 



CHAPTER V 

SPRINGS 

44. Helical Springs. — The most common form of spring used 
in machinery is the spiral or helical spring made of round brass 
or steel wire. Such springs may be used to resist extension or 
compression or they may be used to resist a twisting moment. 

Tension and Compression 
Let L = length of axis of spring 
D = mean diameter of spring 
I = developed length of wire 
d = diameter of wire 

R = ratio -r 
d 

n = number of coils 
P = tensile or compressive force 
X = corresponding extension or compression 
S = safe torsional or shearing strength of wire 

= 45,000 to 60,000 for spring brass wire 

= 75,000 to 115,000 for cast steel, tempered 
G = modulus of torsional elasticity 

= 6,000,000 for spring brass wire 

= 12,000,000 to 15,000,000 for cast steel, tempered. 

Then l = \A:^DVTl^ 

If the spring were extended until the wire became straight it 
would then be twisted n times, or through an angle = 27rn and 
the stretch would be Z — L. 

The angle of torsion for a stretch = a; is then 

^ 27:nx f . 

Suppose that a force F' acting at a radius — will twist this 



2 



107 



108 MACHINE DESIGN 

same piece of wire through an angle 6 causing a stress ;S at the 
surface of the wire. Then will the distortion of the surface of the 

wire per inch of length be s = -^ and the stress 

In thus twisting the wire the force required will vary uniformly 
from at the beginning to P' at the end provided the elastic 
limit is not passed, and the average force will be 

P' . P'Dd 

= -^ The work done is therefore — -. — 
2 4 

If the wire is twisted through the same angle by the gradual 
application of the direct pressure P, compressing or extending 
the spring the amount x, the work done will be 

Px ^ P'Dd Px 

• • ^ DO ^^ 

Substituting this value of P' in (c) and solving for x: 

_ GdW^ 
^~10.2Pl 

Substituting the value of 6 from (a) and again solving for x: 

'''- Gd* \ 27:n i ^^' 

If we neglect the original obliquity of the wire then l = 7tDn 
and L = o and equation (e) reduces to 

2.55PW ,.„, 

^ = —g^ (58) 

Making the same approximation in equation (d) we have 

P'=P 

i.e. — a force P will twist the wire through approximately the 
same angle when applied to extend or compress the spring, as if 



HELICAL SPRINGS 109 

applied directly to twist a piece of straight wire of the same 

material with a lever arm = — 

This may be easily shown by a model. 

The safe working load may be found by solving for F' in (b) 
and remembering that F = F' 

2.55D 2.55R ^ ^ 

when >S is the safe shearing strength. 

Substituting this value of F in (58) we have for the safe 
deflection : 

IDS IRS ,„_, 

45. Square Wire. — The value of the stress for a square section is : 

„ 4.247^ 

wher^ d is the side of square. 

The distortion at the corners caused by twisting through an 
angle 6 is: 

dd^ 

'~W2 
Equation (c) then becomes: 

^F'Dl 
2d'd 
The three principal equations (58) , (59) and (60) then reduce to: 

1.5PZD2 
^=^^ (61) 

Sd^ 

. = ^ (63) 

GdV2 ^ ^ 

The square section is not so economical of material as the 
round. 

46. Experiments. — Tests made on about 1700 tempered steel 
springs at the French Spring Works in Pittsburg were reported 
in 1901 by Mr. R. A. French.^ These were all compression springs 

1 Trans. A. S. M. E., Vol. XXIII. 



110 



MACHINE DESIGN 



of round steel and were given a permanent set before testing by- 
being closed coil to coil several times. Table XXXIII gives 
results of these experiments. 



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I I 



HELICAL SPRINGS ^ 111 

The apparent variation of G in the experiments is probably 
due to differences in the quality of steel and to the fact that the 
formula for G in the case of helical springs is an approximate one. 

The same may be said of the values of S, but if these values are 
used in designing similar springs one error will off-set the other. 

In some few cases, as in No. 18, it was necessary to use an 
abnormally high value of S to meet the conditions. This neces- 
sitated a special grade of steel, and great care in manufacture. 
Such a spring is not safe when subjected to sudden and heavy 
loads, or to rapid vibrations, as it would soon break under such 
treatment; if merely subjected to normal stress, it would last for 
years. 

Springs of a small diameter may safely be subjected to a higher 
stress than those of a larger diameter, the size of bar being the 
same. The safe variation of S with R cannot yet be stated. 

There is an important limit which should be here mentioned. 
Springs having two small a diameter as compared with size of bar 
are subjected to so much internal stress in coiling as to weaken 
the steel. A spring, to give good service, should never have R less 
than 3. 

The size of bar has much to do with the safe value of S; the 
probable explanation is this: A large bar has to be heated to a 
higher temperature in working it, and in high carbon steel this 
may cause deterioration; when tempered, the bath does not affect 
it so uniformly, as may be seen by examining the fracture of a 
large bar. 

The above facts must always be taken into consideration in 
designing a spring, whatever the grade of steel used. A safe 
value of S can be determined only by one having an accurate 
knowledge of the physical characteristics of the steel, the pro- 
portions of the spring, and the conditions of use. 

For a good grade of steel the values of aS on p. 112 have been 
found safe under ordinary conditions of service, the value of G 
being taken as 14,500,000. 

For bars over 1^ in. in diameter a stress of more than 100,000 
should not be used. Where a spring is subjected to sudden 
shocks a smaller value of S is necessary. 

As has been noted, the springs referred to in this paper were 
all compression springs. Experience has shown that in close 



112 



MACHINE DESIGN 



coil or extension springs the value of G is the same, but that the 
safe value of S is only about two-thirds that for a compression 
spring of the same dimensions. 

VALUES OF S 





72 = 3 


R = S 


d = f in, or less 

d=^^ in. to 1 in 

d = \^ in. to 1\ in 


112,000 
110,000 
105,000 


85,000 
80,000 
75,000 



47. Spring in Torsion.— If a helical spring is used to resist 
torsion instead of tension or compression, the wire itself is 
subjected to a bending moment. We will use the same notation 
as in the last article, only that P will be taken as a force acting 

tangentially to the circumference of the spring at a distance — 

from the axis, and S will now be the safe transverse strength of 
the wire, having the following values: 
S = 60,000 for spring brass wire 

= 90,000 to 125,000 for cast steel tempered 
E = 9,000,000 for spring brass wire 
= 30,000,000 for cast steel tempered. 
Let 6 = angle through which the spring is turned by P. 
The bending moment on the wire will be the same throughout 

PD . : 

and = -^- This is best illustrated by a model. 

To entirely straighten the wire by unwinding the spring would 
require the same force as to bend straight wire to the curvature 
of the helix. 

To simplify the equations we will disregard the obliquity of 
the helix, then will l = 7iDn and the radius of curvature 

_D 
~ 2 
Let M = bending moment caused by entirely straightening 
the wire; then by mechanics 

,^ EI 2EI 



HELICAL SPRINGS 113 

and the corresponding angle through which spring is turned 
is 27tn. 

But it is assumed that a force P with a radius -^ turns the 

spring through an angle 6. 

PD 2EI 6 



Solving for 6 : 
and if wire is round 



D 27in 
Eld ^Eld 
nDn I 



e=^j (a) 



„ 10.2PDI ,_,, 



The bending moment for round wire will be 

PD Sd^ 



(65) 



2 10.2 

and this will also be the safe twisting moment that can be 
applied to the spring when /S = working strength of wire. The 
safe angle of deflection is found by substituting this value of 

™in(64): 

21S 
Reducing: 6=^=^- (66) 

£j(i 

48. Flat Springs. — Ordinary flat springs of uniform rectangular 
cross-section can be treated as beams and their strength and 
deflection calculated by the usual formulas. 

In such a spring the bending and the stress are greatest at 
some one point and the curvature is not uniform. 

To correct this fault the spring is made of a constant depth 
but varying width. 

If the spring is fixed at one end and loaded at the other the 
plan should be a triangle with the apex at loaded end. If it is 
supported at the two ends and loaded at the center, the plan 
should be two triangles with their bases together under the load 
forming a rhombus. The deflection of such a spring is one and 
a half times that of a rectangular spring. 



114 MACHINE DESIGN 

As such a spring might be of an inconvenient width, a com- 
pound or leaf-spring is made by cutting the triangular spring 
into strips parallel to the axis, and piling one above another as 
in Fig. 37. 

This arrangement does not change the principle, save that the 
friction between the leaves may increase the resistance somewhat. 





i 



I 



Fig. 37. 

LetJ = length of span 

6 = breadth of leaves 

^ = thickness of leaves 

n = number of leaves 
W = load at center 
A = deflection at center. 

S and E may be taken as 80,000 and 30,000,000 respectively. 
Strength: 

_ Wl Snhf 

(67) 
Elasticity: 



lf± — 


4 6 




w= 


2 Snbt^ 
3* I 




32EI 
• A 


where / = 


nht^ 
12 



SEnbfi ^^^^ 

49. Elliptic and Semi-elliptic Springs. — Springs as they are usu- 
ally designed for service differ in some respects from those just 
described, as may be seen by reference to Fig. 38. A band is used 



ELLIPTIC SPRINGS 



115 



at the center to confine the leaves in place. As this band con- 
strains the spring at the center it is best to consider the latter as 

I — w 
made up of two cantilevers each having a length of where w 

is the width of band. The spring usually contains several full- 
length leaves with blunt ends, the remaining leaves being 
graduated as to length and pointed as in Fig. 38. The blunt 
full-length leaves constitute a cantilever of uniform cross-section, 
while the graduated leaves form a cantilever of uniform strength. 
Under similar conditions as to load and fiber stress the latter 
will have a deflection 50 per cent greater than the former. Sup- 




FiG. 38. 

posing that there is no initial stress between the leaves caused 
by the band, both sets must have the same deflection. This 
means that more than its proportion of the load will be carried 
by the full-length set and consequently it will have a greater 
fiber stress. This difiiculty can be obviated by having an initial 
gap between the graduated set and the full-length set and 
closing this with the band. 

If this gap is made half the working deflection of the spring, 
the total deflection of the graduated set under the working load 
will be 50 per cent greater than that of the full-length set and the 
fiber stress will be uniform. 

The load will then be divided between the two sets in propor- 
tion to the number of leaves in each. 

One of the full-length leaves must be counted as a part of the 
graduated set. When the gap is closed by a band there will be 
an initial pull on the band due to the deflection of the spring. 

This can be determined for any given spring by regarding the 
two sets of leaves as simple beams the sum of whose deflections 
under the pull P is equal to the depth of the gap. 

Full elliptic springs can be designed in a similar manner but the 
total deflection will be double that of the semi-elliptic spring. 



116 MACHINE DESIGN 

The mathematical discussion of which the following is an 
abstract was given by Mr. E. R. Morrison/ who, as far as the 
author knows, was the first to treat the subject in this way. 

Let c = whole length of spring 
w = width of band 

c — w 
I = — ^^ = length of each cantilever 

b = breadth of leaves 

t = thickness of one leaf 

n = total number of leaves 

n' = number of full-length leaves 

n'' = number of graduated leaves 

, . n' 
r = ratio — 
n 

S = maximum fiber stress in spring 

aS' = maximum fiber stress in full-length leaves 

A = total deflection of spring 

A' = total deflection of full-length leaves if unbanded 

A" = total deflection of graduated leaves if unbanded 

P = total load on spring 

P' = portion of load on one end of full-length leaves 

P" = portion of load on one end of graduated leaves. 

Assuming that the maximum stress should be the same in both 
parts: 

QP'l_ QP'n 

' ' ri/bi^~nrb¥ 
and 

P" n"' ^^^ 

The deflections, as already stated on preceding page, will be 

unequal. For a cantilever of uniform section (full-length leaves) : 

4P'Z3 
A'= Cb) 

and for one of uniform strength (graduated leaves) : 

(\P''P 
,^ or i 

En"ht^ ^^ 

» Mchy., N. Y., Jan., 1910. 



ELLIPTIC SPRINGS 117 



But from (a) 



and 

and 
But also 



and A"-A'=-=^^. (d) 

Equation (d) gives the excess of tht deflection of the graduated 
portion over that of the full-length portion and is the proper 
depth of gap between the two portions before banding. To 
find the effect of the banding: 

Let P6= force exerted by band 

d' = deflection caused by band in full-length leaves 
d^' = deflection caused by band in graduated leaves. 



P' 


p/, 


n' 


n" 


A' 


4p,./3 


En''ht\ 


A': 


2P'n^ 


En^'bt^ 


P" 


P 


n" 


2n 


A'- 


Pl^ 



Then, 
and 



d'=^^ (e) 

^ En"W ^^ 

(Since force at end of each cantilever = -^.) 
By division and cancellation, 

P/3 

The depth of gap = d' + (i'' = from equation (d). 



Combining : 



d"+l?^V=^' 



d 



3n' Enbf 

3n' \ Pl^ 



\3n' + 2W Enbt^ ^ ^ 



118 MACHINE DESIGN 

Equating (f) and (h) : 

3Pbl^ / 3n' \ PI' 



3n' \ 



En"U^ \Zn' + 2n") EnW 
Solving for P^ : 

p^^ '^''^" p 
Or letting n^ =rn n" = (l-r)n 

p»-'^p (i) 

2 + r 

and this equation gives the force exerted by the band in terms 
of the total load. 

The woiking deflection of the spring may be obtained as 
follows: 

The total deflection of the graduated leaves undei the load P 
is by equation (c) : 

. „_^P^ ^ 3PZ^ 
~ En''bt^~ EnU^' 

But a part of this total is produced by the banding, equation (h) : 

Sn' \ Pl^ 



\3n' + 2n'7 



+ 2n'7 Enbt^ 

The remaining deflection or that due to the application of the 
load P is: 

PP 



^"-'"-('-3n^) 



Enht^ 



6 Pl^ 

^=7+2' EnM^- ^^ 

But 

Snht^ 
P = 2(P' + Pn=2^^ 

where P'+P^'^load at each end of spring and (P'+P'0^ = 
bending moment at band. 
Substituting in (j) and reducing: 

If all the leaves are full length : 

r = Und A =-3^. 



ELLIPTIC SPRINGS 119 

If all the leaves are graduated: 

r = and A = -r=r-' 
Et 

PROBLEMS 

1. A spring balance is to weigh 50 lb. with an extension of 2 in., the 
diameter of spring being f in. and the material, tempered steel. 

Determine the diameter and length of wire, and number of coils. 

2. Determine the safe twisting moment and angle of torsion for the spring 
in example 1, if used for a torsional spring. 

3. Test values of G and S from data given in Table XXXIII. 

4. By using above table design a spring 8 in. long to carry a load of 2 tons 
without closing the coils more than half way. 

5. Design a compound flat spring for a locomotive to sustain a load of 
16,000 lb. at the center, the span being 40 in., the number of leaves 12 and 
the material steel. 

6. Determine the maximum deflection of the above spring, under the 
working load. 

7. A semi-elliptic spring has 9 leaves in all and 6 graduated leaves, and 
the load on each end is P=4000 lb. Develop formulas for the fiber stress 
in each set of leaves if there is no initial stress. Determine proper breadth 
and thickness of leaves if length of span is 42 in. 

8. In Prob. 7 develop a formula for the necessary gap to equalize the 
fiber stresses. 

9. In Prob. 8 determine the pull on the band due to the initial stress. 

10. A semi-elliptic spring has 4 leaves 36 in. long, and 12 graduated 
leaves. The leaves are all 4 in. wide and | in. thick, and the band at the 
center is 4 in. wide. If there is no initial stress find the share of the load 
and the fiber stress on each set of leaves when there is a load of 6 tons on 
the center. Also determine deflection. 

11. In Prob. 10, determine the amount of gap needed to equalize the 
stresses in the two sets of leaves, and the pull on the band at the center. 
Determine the deflection under the load. 

12. Measure various indicator springs and determine value of G from 
rating of springs. 

13. Measure various brass extension springs, calculate safe static load 
and safe stretch. 

14. Make an experiment on torsion spring to determine distortion under 
a given load and calculate value of E. 

REFERENCES 

Vibration of Springs. Am. Mach., May 11, 1905. 

Tables of Loads and Deflections. Am. Mach., Dec. 20, 1906. 

The Constructor. Reuleavx. 



' CHAPTER VI 
SLIDING BEARINGS 

50. Slides in General. — The surfaces of all slides should have 
sufficient area to limit the intensity of piessure and prevent 
forcing out of the^ lubricant. No general rule can be given for 
the limit of pressure. Tool marks parallel to the sliding motion 
should not be allowed, as they tend to start grooving. The 
sliding piece should be as long as practicable to avoid local wear 
on stationary piece and for the same reason should have sufficient 
stiffness to prevent springing. A slide which is in continuous 
motion should lap over the guides at the ends of stroke, to prevent 
the wearing of shoulders on the latter and the finished surfaces 
of all slides should have exactly the same width as the surfaces 
on which they move for a similar reason. 

Where there are two parallel guides to motion as in a lathe or 
planer it is better to have but one of these depended upon as an 
accurate guide and to use the other merely as a support. It 
must be remembered that any sliding bearing is but a copy of 
the ways of the machine on which it was planed or ground and in 
turn may reproduce these same errors in other machines. The 
interposition of hand-sci aping is the only cure for these hereditary 
complaints. 

In designing a slide one must consider whether it is accuracy 
of motion that is sought, as in the ways of a planer or lathe, or 
accuracy of position as in the head of a milling machine. Slides 
may be divided according to their shapes into angular, flat and 
circular slides. 

51. Angular Slides. — An angular slide is one in which the 
guiding surface is not normal to the direction of pressure. There 
i-s a tendency to displacement sideways, which necessitates a 
second guiding surface inclined to the first. This oblique 
pressure constitutes the principal disadvantage of angular slides. 

120 



ANGULAR SLIDES 



121 



Their principal advantage is the fact that they are either self- 
adjusting for wear, as in the ways of lathes and planers, or 
require at most but one adjustment. 

Fig. 39 shows one of the Vs of an ordinary planing machine. 
The platen is held in place by gravity. The angle between the 
two surfaces is usually 90 degrees but may be more in heavy 
machines. The grooves g, g are intended to hold the oil in place; 
oiling is sometimes effected by small rolls recessed into the lower 
piece and held against the platen by springs. 

The principal advantage of this form of way is its ability to 
hold oil and the great disadvantage, its faculty for catching chips 
and dirt. 

Fig. 40 shows an inverted V such as is common on the ways of 
engine lathes. The angle is about the same as in the preceding 
form but the top of the V should be rounded as a precaution 
against nicks and bruises. 





Fig. 39. 



Fig. 40. 



The inverted V is preferred for lathes since it will not catch 
dirt and chips. It needs frequent lubrication as the oil runs off 
rapidly. Some lathe carriages are provided with extensions 
filled with oily felt or waste to protect the ways from dirt and 
keep them wiped and oiled. Side pressure tends to throw the 
carriage from the ways; this action may be prevented by a heavy 
weight hung on the carriage or by gibbing the carriage at the 
back (see Fig. 46). The objection to this latter form of con- 
struction is the fact that it is practically impossible to make 
and keep the two T's and the gibbed slide all parallel. 

Fig. 41 shows a compound V sometimes used on heavy ma- 
chines. The obtuse angle (about 150 degrees) takes the heavy 



122 



MACHINE DESIGN 



vertical pressure, while the sides, inclined only 8 or 10 degrees, 
take any side pressure which may develop. 

52. Gibbed Slides. — All slides which are not self-adjusting for 
wear must be provided with gibs and adjusting screws. Fig. 42 
shows the most common form as used in tool slides for lathes and 
planing machines. 





Fig. 41. 



Fig. 42. 



The angle employed is usually 60 degrees; notice that the 
corners c c are clipped for strength and to avoid a corner bearing; 
notice also the shape of gib. It is better to have the points of 
screws coned to fit gib and not to have flat points fitting recesses 
in gib. The latter form tends to spread the joint apait by 

forcing the gib down. If the gib is 

too thin it will spring under the screws 

and cause uneven wear. 

/W The cast-iron gib, Fig. 43, is free 

from this latter defect but makes 

the slide rather clumsy. The screws, 

Pjq 43 however, are more accessible in this 

form. Gibs are sometimes made 

slightly tapering and adjusted by a screw and nut giving endwise 

motion. 



I 



53. Flat Slides. — This type of slide requires adjustment in 
tw^o directions and is usually provided with gibs and adjusting 
screws. Flat ways on machine tools are the rule in English 
piactice and aie gradually coming into use in this country. 
Although more expensive at first and not so simple they are 
more durable and usually more accurate than the angular ways. 

Fig. 44 illustrates a flat way for a planing machine. The other 



FLAT SLIDES 



123 



way would be similar to this but without adjustment. The 
normal pressure and the friction are less than with angular ways 
and no amount of side pressure will lift the platen from its 
position. 




Fig. 44. 



Fig. 45. 



Fig. 45 shows a portion of the ram of a shaping machine and 
illustrates the use of an L gib for adjustment in two directions. 
Fig. 46 shows a gibbed slide for holding down the back of a lathe 
carriage with two adjustments. 





Fig. 46. 



Fig. 47. 



The gib g is tapered and adjusted by a screw and nuts. The 
saddle of a planing machine or the table of a shaper usually has 
a rectangular gibbed slide above and a taper slide below, this 
form of the upper slide being necessary to hold the weight of the 



124 



MACHINE DESIGN 



overhanging metal (see Fig. 47). Some lathes and planers are 
built with one V or angular way for guiding the carriage or 
platen and one flat way acting merely as a support. 

54. Circular Guides. — Examples of this form may be found in 

^ the column of the drill press 

and the overhanging arm of 

the milling machine. The 

cross heads of steam engines 

are sometimes fitted with 

circular guides; they are more 

frequently flat or angular. 

One advantage of the circular 

form is the fact that the cross 

head can adjust itself to bring 

the wrist pin parallel to the 

crank pin. The guides can be bored at the same setting as 

the cylinder in small engines and thus secure good alignment. 

Fig. 48 illustrates various shapes of cross head slides in common 

use. 






Fig. 48. 



55. Stuffing Boxes. — In steam engines and pumps the glands 
for holding the steam and water packing are the sliding bearings 




Fig. 49. 



which cause the greatest friction and the most trouble. Fig. 49 
shows the general arrangement. B is the stuffing box attached 
to the cylinder head; R is the piston rod; G the gland adjusted by 



STUFFING BOX 125 

nuts on the studs shown; P the packing contained in a recess in 
the box and consisting of rings, either of some elastic fibrous 
material like hemp and woven rubber cloth or of some soft metal 
like Babbitt metal. The pressure between the packing and the 
rod, necessary to prevent leakage of steam or water, is the cause 
of considerable friction and lost work. Experiments made from 
time to time in the laboiatories of the Case School have shown 
the extent and manner of variation of this friction. The results 
for steam packings may be summarized as follows: 

1. That the softer rubber and graphite packings, which are self- 
adjusting and self -lubricating, as in Nos. 2, 3, 7, 8, and 11, con- 
sume less power than the harder varieties. No. 17, the old 
braided flax style, gives very good results. (See Table 
(XXXIV.) 

2. That oiling the rod will reduce the friction with any packing. 

3. That there is almost no limit to the loss caused by the 
injudicious use of the monkey-wrench. 

4. That the power loss varies almost directly with the steam 
pressure in the harder varieties, while it is approximately con- 
stant with the softer kinds. 

The diameter of rod used — 2 in. — would be appropriate for 
engines from 50 to 100 horse-power. The piston speed was about 
140 ft. per minute in the expeiiments, and the horse-power varied 
from .036 to .400 at 50 lb. steam piessure, with a safe average 
for the softer class of packings of .07 horse-power. 

At a piston speed of 600 ft. per minute, the same friction would 
give a loss of from .154 to 1.71 with a working average of .30 
horse-power, at a mean steam pressure of 50 lb. 

In Table XXXIV Nos. 6, 14, 15, and 16 are square, hard rubber 
packings without lubricants. 

Similar experiments on hydraulic packings under a water 
pressure varying from 10 to 80 lb. per square inch gave results 
as shown in Table XXXVI. 

The figures given are for a 2-in. rod running at an average 
piston speed of 140 ft. per minute. 



V. 



126 



MACHINE DESIGN 



TABLE XXXIV 









Average 


Horse- 




Kind of 
packing 


No 
trials 


Total 

time of 

run in 

minutes 


horse- 
power 
con- 
sumed 
by each 
box 


power 
con- 
sumed 
at 50 
lb. pres- 
sure 


Remarks on leakage, etc. 


1 


5 


22 


.091 


.085 


Moderate leakage. 


2 


8 


40 


.049 


.048 


Easily adjusted; slight leakage. 


3 


5 


25 


.037 


.036 


Considerable leakage. 


4 


5 


25 


.159 


.176 


Leaked badly. 


5 


5 


25 


.095 


.081 


Oiling necessary; leaked badly. 


6 


5 


25 


.368 


.400 


Moderate leakage. 


7 


5 


25 


.067 


.067 


Easily adjusted and no leakage. 


8 


5 


25 


.082 


.082 


Very satisfactory; slight leakage. 


9 


3 


15 


.200 


.182 


Moderate leakage. 


10 


3 




.275 




Excessive leakage. 


11 


5 


25 


.157 


.172 


Moderate leakage. 


12 


5 


25 


.266 


.330 


Moderate leakage. 


13 


5 


25 


.162 


.230 


No leakage; oiling necessary. 


14 


5 


25 


.176 


.276 


Moderate leakage; oiling necessary 


15 


5 


25 


.233 


.255 


Difficult to adjust; no leakage. 


16 


5 


25 


.292 


.210 


Oiling necessary; no leakage. 


17 


5 


25 


.128 


.084 


No leakage. 



TABLE XXXV 



Kind of 
packing 


Horse-power consumed by each box, when pressure was 
applied to gland nuts by a 7-in. wrench 


Horse-power 

before and after 

oiling rod 




5 1b. 


8 1b. 


10 1b. 


12 1b. 


14 1b. 


16 1b. 


Dry 


Oiled 


1 

3 

4 

5 

6 

7 

8 

9 

11 

12 

13 

15 

16 

17 


.120 




.136 


















.055 
.154 


.021 
.123 




.248 
.220 
.348 
.126 
.363 
.666 
.405 
.161 
.317 
.526 
.327 
.198 




.303 




.390 


.430 
.228 
.500 








.323 
.067 
.533 
.666 
.454 
.454 


.194 
.053 
.236 
.636 
.176 
.122 


.260 
.535 


.330 
.520 


.340 
.533 


.454 
.242 
.394 








.359 

.582 


.454 












.860 

.277 












.380 



















STUFFING BOXES 



127 



TABLE XXXVI 



No. of 


Av. H. P. Av 


. H. P. 


Max, 


Min. 


Av. H. P. 

for entire 

test 


packing 


at 20 lb. at 70 lb. 


H. P. 


H.P. 


1 


.077 


351 


.452 


.024 


.259 


2 


.422 


500 


.512 


.167 


.410 


3 


.130 


178 


.276 


.035 


.120 


4 


.184 


195 


.230 


.142 


.188 


5 


.146 


162 


.285 


.069 


.158 


6 


.240 


200 


.255 


.071 


.186 


7 


.127 


192 


.213 


.095 


.154 


8 


.153 


174 


.238 


.112 


.165 


9 


.287 


469 


.535 


.159 


.389 


10 


.151 


160 


.226 


.035 


.103 


11 


.141 


156 


.380 


.064 


.177 


12 


.053 


095 


.143 


.035 


.090 



Packings Nos. 5, 6, 10 and 12 are braided flax with graphite 
lubrication and are best adapted for low pressures. Packings 
Nos. 3, 4 and 7 are similar to the above but have paraffine lubri- 
cation. Packings Nos. 2 and 9 are square duck without lubri- 
cant and are only suitable for very high pressures, the friction 
loss being approximately constant. 



PROBLEMS 

Make a careful study and sketch of the sliding bearings on each of the 
following machines and analyze as to (a) Purpose. (6) Character, (c) 
Adjustment, (d) Lubrication. 

1. An engine lathe. 

2. A planing machine. 

3. A shaping machine. 

4. A milling machine. 

5. An upright drill. 

6. A Corliss engine. 

7. A locomotive engine. 

8. A gas-engine. 

9. An air-compressor. 



CHAPTER VII 
JOURNALS, PIVOTS AND BEARINGS 

56. Journals. — A journal is that part of a rotating shaft which 
rests in the bearings and is of necessity a surface of revolution, 
usually cylindrical or conical. The material of the journal is 
generally steel, sometimes soft and sometimes hardened and 
ground. 

The material of the bearing should be softer than the journal 
and of such a quality as to hold oil readily. The cast metals 
such as cast iron, bronze and Babbitt metal are suitable on 
account of their poious, granular character. Wood, having the 
grain normal to the bearing surface, is used where water is the 
lubiicant, as in water wheel steps and stern beaiings of propellers. 

Bearing materials may naturally be divided into soft and hard 
metals. The standard soft metal is so-called ''genuine Babbitt," 
of the following composition: 

Tin, 85 to 89 per cent. 
Copper, 2 to 5 per cent. 
Antimony, 7 to 10 pei cent. 

The substitution of lead for tin and the omission of the copper 
makes a cheaper and softer metal suitable for low pressures and 
speeds. The addition of more antimony hardens the metal. 

The hard metals include the various brasses and bronzes 
ranging from soft yellow brass to phosphor and aluminum 
bronzes. 

Professor R. C. Carpenter recommends as suitable for a bearing 
an aluminum bronze whose composition is: 

Aluminum, 50 per cent. 
Zinc, 25 per cent. 
Tin, 25 per cent. 

This metal is light, fairly hard, and will not melt readily.^ 

1 Trans. A. S. M. E., Vol. XXVII, p. 425. 

128 



JOURNAL BEARINGS 



129 



57. Adjustment. — Bearings wear more or less lapidly with use 
and need to be adjusted to compensate for the wear. The 
adjustment must be of such a character and in such a direction 
as to take up the wear and at the same time maintain as far as 
possible the correct shape of the bearing. The adjustment 
should then be in the line of the greatest pressure. 

Fig. 50 illustrates some of the more common ways of adjusting 
a bearing, the arrows showing the direction of adjustment and 
presumably the direction of pressure; (a) is the most usual where 
the principal wear is vertical, (d) is a form frequently used on 
the main journals of engines when the wear is in two directions, 



O 



^X\' 






XX 



d e 

Fig. 50. 







A 




Fig. 51. 



horizontal on account of the steam pressure and vertical on 
account of the weight of shaft and fly-wheel. All of these are 
more or less imperfect since the bearing, after wear and adjust- 
ment, is no longer cylindrical but is made up of two or more 
approximately cylindrical surfaces. 

A bearing slightly conical and adjusted endwise as it wears, is 
probably the closest approximation to correct practice. 

Fig. 51 shows the main bearing of the Porter- Allen engine, 
one of the best examples of a four part adjustment. The cap is 
adjusted in the normal way with bolts and nuts; the bottom can 
be raised and lowered by liners placed underneath; the cheeks 
can be moved in or out by means of the wedges shown. Thus 
it is possible not only to adjust the bearing for wear but to align 
the shaft perfectly. 

A three part bearing for the main journal of an engine is 



130 



MACHINE DESIGN 



shown in Fig. 52. In this bearing there is one horizontal adjust- 
ment instead of two as in Fig. 51. 

The main bearing of the spindle in a lathe, as shown in Fig. 53, 
offers a good example of symmetrical adjustment. The head- 





FiG. 52. 



Fig. 53. 



stock A has a conical hole to receive the bearing B, which latter 
can be moved lengthwise by the nuts FG. The bearing may be 
split into two, three or four segments or it may be cut as shown 
in e, Fig. 50, and sprung into adjustment. A careful distinction 




Fig. 54. 



must be made between this class of bearing and that before 
mentioned, where the journal itself is conical and adjusted end- 
wise. A good example of the latter form is seen in the spindles 
of many milling machines. 

Fig. 54 shows the spindle of an engine lathe complete with its 
two bearings. The end thrust is taken by a fiber washer backed 



JOURNAL BEARINGS 



131 



by an adjusting collar and check nut. Both bearings belong to 
the class shown in Fig. 53. 

A conical journal with end adjustment is illustrated in Fig. 55, 
which shows the spindle of a milling machine. The frOnt journal 
is conical and is adjusted for wear by drawing it back into its 
bearing with the nut. The rear journal on the other hand is 
cylindrical and its bearing is adjusted as are those just described. 
The end thrust is taken by two loose rings at the front end of the 
spindler 




Fig. 55. 



58. Lubrication. — The bearings of machines which run inter- 
mittently, like most machine tools, are oiled by means of simple 
oil holes, but machinery which is in continuous motion as is the 
case with line shafting and engines requires some automatic 
system of lubrication. There is not space in this book for a 
detailed description of all the various types of oiling devices and 
only a general classification will be attempted. 

Lubrication is effected in the following ways: 

1. By grease cups. 

2. By oil cups. 

3. By oily pads of felt or waste. 

4. By oil wells with rings or chains for lifting the oil. 

5. By centrifugal force through a hole in the journal itself. 
Grease cups have little to recommend them except as auxiliary 

safety devices. Oil cups are various in their shapes and methods 
of operation and constitute the cheap class of lubricating devices. 
They may be divided according to their operation into wick oilers, 
needle feed, and sight feed. The two first mentioned are nearly 
obsolete and the sight-feed oil cup, which drops the oil at regular 



132 



MACHINE DESIGN 




intervals through a glass tube in plain sight, is in common use. 
The best sight-feed oiler is that which can be readily adjusted 
as to time intervals, which can be turned on or off without dis- 
turbing the adjustment and which shows clearly by its appear- 
ance whether it is turned on. On engines and 
electric machinery which are in continuous use 
day and night, it is very important that the 
oiler itself should be stationary, so that it 
may be filled without stopping the machinery. 
A modern sight-feed oiler for an engine is 
illustrated in Fig. 56. T is the glass tube where 
the oil drop is seen. The feed is regulated by 
the nut A^, while the lever L shuts off the 
oil. Where the lever is as shown the oil is 
dropping, when horizontal the oil is shut off. 

The nut can be adjusted once for all, and the 
position of the lever shows immediately whether 
or not the cup is in use. 

In modern engines particular attention has 
been paid to the problem of continuous oiling. 
The oil cups are all stationary and various 
ingenious devices are used for catching the drops of oil from 
the cups and distributing them to the bearing surfaces. For 
continuous oiling of stationary bearings, as in line shafting 
and electric machinery, an oil well below the bearing is preferred, 
with some automatic means of pumping 
the oil over the bearing, when it runs back 
by gravity into the well. Porous wicks and 
pads acting by capillary attraction are un- 
certain in their action and liable to become 
clogged. For bearings of medium size, 
one or more light steel rings running loose 
on the shaft and dipping into the oil, as 
shown in Fig. 57, are the best. For large 
bearings flexible chains are employed 
which take up less room than the ring. 

Cases have been reported, however, where suction oilers on 
line shafting have proved more efficient than ring oilers. One 
instance is quoted where a suction oiler has been in continuous 



Fig. 56. 




Fig. 57. 



LUBRICATION 



133 



use for nearly thirty years and has worked perfectly during 
that entire period/ Much depends on the care of such devices, 
the prevalence or absence of dust, and the quality of oil used. 
Centrifugal oilers are most used on parts which cannot readily 




B 



i 



\.-,X\ 



6 



C 



Fig. 58. 



be oiled when in motion, such as loose pulleys and the crank 
pins of engines. 

Fig. 58 shows two such devices as applied to an engine. In 
A the oil is supplied by the waste from the main journal; in B 
an external sight-feed oil cup is used which supplies oil to the 
central revolving cup C. 




Fig. 59. 

Loose pulleys or pulleys running on stationary studs are best 
oiled from a hole running along the axis of the shaft and thence 
out radially to the surface of the bearing. See Fig. 59. A loose 
bushing of some soft metal perforated with holes is a good safety 
device for loose pulleys. 

» Trans. A. S. M. E., Vol. XXVII, p. 488. 



134 MACHINE DESIGN 

Note: For adjustable pedestal and hanging bearings see the 
chapter on shafting. 

59. Friction of Journals : 

Let TF = the total load of a journal in pounds 
Z = the length of journal in inches 
d = th.e diameter of journal in inches 
A^ = number of revolutions per minute 
-?;= velocity of rubbing in feet per minute 
F=friction at surface of journal in pounds 
= W tan ¥ nearly, where ¥ is the angle of repose for 
the two materials. 

If a journal is properly fitted in its bearing and does not 
bind, the value of F will not exceed W tan ¥ and may be slightly 
less. The value of tan ¥ varies according to the materials used 
and the kind of lubrication, from .05 to .01 or even less. See 
experiments described in Art. 62. The work absorbed in friction 
may be thus expressed: 

■n TTTj^ irr ^^N 7zdNWtan¥ .^ ,- . ,^^^ 

Fv=Wtan¥x-Tw- = tt^ ft. lbs. per mm. (70) 

60. Limits of Pressure. — Too great an intensity of pressure 
between the surface of a journal and its bearing will force out 
the lubricant and cause heating and possibly "seizing." The 
safe limit of pressure depends on the kind of lubricant, the 
manner of its application and upon whether the pressure is con- 
tinuous or intermittent. The projected area of a journal, or the 
product of its length by its diameter, is used as a divisor. 

The journals of railway cars offer a good example of con- 
tinuous pressure and severe service. A limit of 300 lb. per 
square inch of projected area has been generally adopted in such 
cases. 

In the crank and wrist pins of engines, the reversal of pressure 
diminishes the chances of the lubricant being squeezed out, and 
a pressure of 500 lb. per square inch is generally allowed. 

The use of heavy oils or of an oil bath, and the employment 
of harder materials for the journal and its bearing allow of even 
greater pressures. 



BEARING PRESSURES 135 

Professor Barr's investigations of steam engine proportions^ 
show that the pressure per square inch on the cross-head pin 
varies from ten to twenty times that on the piston, while the 
intensity of pressure on the crank pin is from two to eight times 
that on the piston. Allowing a mean pressure on the piston of 
50 lb. per square inch would give the following range of pressures: 

Minimum Maximum 

Wrist pins 500 1,000 

Crank pins 100 400 

The larger values for the wrist pins are allowable on account 
of the comparatively low velocity of rubbing. Naturally the 
larger values for the pressure are found in the low-speed engines. 

A discussion of the subject of bearings is reported in the trans- 
actions of the American Society of Mechanical Engineers for 
1905-06 and some valuable data are furnished. 

Mr. George M. Basford says that the bearing areas of locomo- 
tive journals are determined chiefly by the possibilities of lubri- 
cation. Crank pins may be loaded to from 1500 to 1700 lb. per 
square inch, since the reciprocation of the rods makes lubrication 
easy. Wrist pins have been loaded as high as 4000 lb. per square 
inch, the limited arc of motion and the alternating pressures 
making this possible. 

Locomotive driving journals on the other hand are limited 
to the following pressures: 

Passenger locomotives 190 lb. per square inch. 

Freight locomotives 200 lb. per square inch. 

Switching locomotives 220 lb. per square inch. 

Cars and tender bearings 300 lb. per square inch. 

Mr. H. G. Reist gives some figures on the practice of the 
General Electric Company, for motors and generators. 

This company allows from 30 to 80 lb. pressure per square 
inch with an average value of from 40 to 45 lb. The rubbing 
speeds vary from 40 ft. to 1200 ft. per minute. Mr. Reist quotes 
approvingly the formula of Dr. Thurston's, viz.: That the 
product of the pressure in pounds per square inch and the 
rubbing speed in feet per minute should not exceed 50,000. 

The ratio of length to diameter of journal is given as about 
3.1 but a smaller ratio is used in special cases. 

' Trans. A. S. M. E., Vol. XVIII. 



136 MACHINE DESIGN 

Oil rings placed not further than 8 in. apart have given good 
results for many years. For bearings more than 1 ft. in diameter, 
a forced circulation of oil is recommended to carry off the heat 
generated. 

The practice of one of the largest firms of Corliss engine builders 
in this country may be summarized as follows: 

All bearings are lined with best Babbitt metal, cast, hammered 
in place and bored. 

Lubrication is effected by pressure oil cups, the oil dropping 
from the cup into a cored pocket in the top shell of the bearing, 
this pocket being filled with waste. 

The speed of the shafts is between 75 and 150 revolutions per 
minute and the allowable pressure on the journal is 140 lb. 
per square inch of projected area. (This is exclusive of steam 
pressure.) 

The bearings of horizontal engines are usually four-part shells 
with the cap separate from the upper shell and the lower shell 
resting on a rib at center, which makes it self-adjustable. 

The bearings of vertical engines are two-part shells. 

A careful reading of the whole discussion will repay any one 
who has to design shaft bearings of any description.^ 

61. Heating of Journals. — The proper length of journals 
depends on the liability of heating. 

The energy or work expended in overcoming friction is con- 
verted into heat and must be conveyed away by the material 
of the rubbing surfaces. If the ratio of this energy to the area 
of the surface exceeds a certain limit, depending on circumstances, 
the heat will not be conveyed away with sufficient rapidity and 
the bearing will heat. 

The area of the rubbing surface is proportional to the projected 
area or product of the length and diameter of the journal, and 
it is this latter area which is used in calculation. 

Adopting the same notation as is used in Art. 59, we have from 
equation (70). 

the work of friction = ^n • ^^- ^t>' P^^ ^^^• 

or = 7idNWtan¥ inch pounds. 
» Trans. A. S. M. E., Vol. XXVII. 



LENGTH OF JOURNALS 137 

The work per square inch of projected area is then: 

TzdNWtanW nNWtanW. 

^= Vd = T (^) 



Solving in (a) for I 



, TzNWtanW 

1 = • (b) 

w 



w . . 

Let — — wr = C a coefficient whose value is to be obtained by 



experiment; then 



C = ^andZ = -^- (71) 



Crank pins of steam . engines have perhaps caused more 
trouble by heating than any other form of journal. A com- 
parison of eight different classes of propellers in the old U. S. Navy 
showed an average value for C of 350,000. 

A similar average for the crank pins of thirteen screw steamers 
in the French Navy gave C = 400,000. 

Locomotive crank pins which are in rapid motion through the 
cool outside air allow a much larger value of (7, sometimes more 
than a million. 

Examination of ten modern stationary engines shows an 
average value of C = 200,000 and an average pressure per square 
inch of projected area =300 lb. 

The investigations of Professor Barr above referred to show a 
wide variation in the constants for the length of crank pins in 

rrp 

stationary engines. He prefers to use the formula : I — K-j- + B 

where K and B are constants and L= length of stroke of engine 
in inches. We may put this in another form since: 

HP WN 

-^^= where W is the total mean pressure. 

The formula then becomes: 



JVN 



138 MACHINE DESIGN 

The value of B was found to be 2.5 in. for high-speed and 2 in. 
for low-speed engines, while K fluctuated from .13 to .46 with 
an average of .30 in the former class, and from .40 to .80 with an 
average of .60 in the low-speed engines. 

If we adopt average values we have the following formulas 
for the crank pins of modern stationary engines : 

WN 
High-speed engines ^ = g^^-^+ 2.5 in. (73) 

WN 
Low-speed engines ^ = 330 qqq +2 m. (74) 

Compare these formulas with (71) when values of C are 
introduced. 

In a discussion on the subject of journal bearings in 1885,^ 
Mr. Geo. H. Babcock said that he had found it practicable to 
allow as high as 1200 lb. per square inch on crank pins while the 
main journal could not carry over 300 lb. per square inch without 
heating. One rule for speed and pressure of journal bearings 
used by a well-known designer of Corliss engines is to multiply 
the square root of the speed in feet per second by the pressure 
per square inch of projected area and limit this product to 350 
for horizontal engines and 500 in vertical engines. 

62. Experiments. — Some tests made on a steel journal 3 J in. 
in diameter and 8 in. long running in a cast-iron bearing and 
lubricated by a sight-feed oiler, will serve to illustrate the friction 
and heating of such journals. 

The two halves of the bearing were forced together by helical 
springs with a total force of 1400 lb., so that there was a pressure 
of 54 lb. per square inch on each half. The surface speed was 
430 ft. per minute and the oil was fed at the rate of about 12 
drops per minute. The lubricant used was a rather heavy 
automobile oil having a specific gravity of 0.925 and a viscosity 
of 174 when compared with water at 20° Cent. 

The length of the run was two hours and the temperature of 
the room 70° fahr. (See Table XXXVII.) 

^ Trans. A. S. M. E., Vol. VI. 



FRICTION OF JOURNALS 



139 



TABLE XXXVII 

Friction of Journal Bearing 



Time 


Rev. per min. 


Temp. fahr. 


Coeff. of friction 


10:03 


500 


69 


.024 


10:15 


482 


82 


.0175 


10:30 


506 


100 


.013 


10:45 


506 


115 


.010 


11:00 


516 


125 


.010 


11:15 
11:30 
11:45 




135 
145 
147 


.004 
.004 
.004 




512 


12:00 


.... 


151 


.007 



Mr. Albert Kingsbury of the Westinghouse Electric and 
Manufacturing Company reports some valuable experiments on 
bearings of unusually large size and under extremely heavy 
pressures.^ 

The bearings were three in number; two, 9 in. in diameter 
and 30 in. long, supporting the shaft, and one 15 in. in diameter 
and 40 in. long to which the pressure was applied. These 
bearings are designated as A, B and C, B being the larger one. 
The bearings were lined with genuine Babbitt metal and scraped 
to fit the shaft. They were flooded with oil from a supply tank 
to which the oil was returned by a pump. 

The runs were usually of about seven hours' duration and 
started with all the parts cool. 

1 Trans. A. S. M. E, Vol. XXVII. . 



140 



MACHINE DESIGN 



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FRICTION OF JOURNALS 



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142 MACHINE DESIGN 

63. Strength and Stiffness of Journals. — A journal is usually 
in the condition of a bracket with a uniform load, and the bending 

Wl 
moment M = — 
2 

Therefore by formula (6) 






s. "^ s 



or d= 1.721 ij^. (75) 



S 

The maximum deflection of such a bracket is 

WP 



A = 



SEI 



7 = "^= "^^' 



d' 



64 8^ A 
64TFP 2.547TF^3 



Stt^A EA 

If as is usual A is allowed to be y^ in., then for stiffness 

or approximately d = 4: V— =-• (77) 

E 

The designer must be guided by circumstances in determining 
whether the journal shall be calculated for wear, for strength 
or for stiffness. A safe way is to design the journal by the 
formulas for heating and wear and then to test for strength and 
deflection. 

Remember that no factor of safety is needed in formula for 
stiffness. 

Note that W in formulas for strength and stiffness is not the 
average but the maximum load. 

64. Caps and Bolts. — The cap of a journal bearing exposed to 
upward pressure is in the condition of a beam supported by the 
holding down bolts and loaded at the center, and may be designed 
either for strength or for stiffness. 



JOURNALS 143 



Let : P = max. upward pressure on cap 
L = distance between bolts 
5= breadth of cap at center 
/i = depth of cap at center 
A = greatest allowable deflection. 

Shh^ PL 



Strength: M 



6 



'=4 



SPL 



Stiffness: A = 



2bS 
4SEI 



(78) 



12 48£'A 



^46^ A 



^-\Z^' (79) 



If A is allowed to be ^^ in. and E for cast iron is taken 
= 18,000,000 

then: /i=.01115L \/^. (80) 

The holding down bolts should be so designed that the bolts 
on one side of the cap may be capable of carrying safely two- 
thirds of the total pressure. 

PROBLEMS 

1. A flat car weighs 20 tons, is designed to carry a load of 40 tons more 
and is supported by two four-wheeled trucks, the axle journals being of 
wrought iron and the wheels 33 in. in diameter. 

Design the journals, considering heating, wear, strength and stiffness, 
assuming a maximum speed of 30 miles an hour, factor of safety = 10 and 
C = 300,000. 

2. The following dimensions are those generally used for the journals of 
freight cars having nominal capacities as indicated: 

Capacity Dimensions of journal 

100,000 lb 4 . 5 by 9 in. 

60,000 lb 4.25 by 8 in. 

40,000 lb 3.75 by 7 in. 



144 MACHINE DESIGN 

Assuming the weight of the car to be 40 per cent of its carrying capacity 
in each instance, determine the pressure per square inch of projected area 
and the value of the constant C {Formula (71) |. 

3. Measure the crank pin of any modern engine which is accessible, 
calculate the various constants and compare them with those given in this 
chapter. 

4. Design a crank pin for an engine under the following conditions: 

Diameter of piston =28 in. 

Maximum steam pressure =90 lb. per square inch. 

Mean steam pressure =40 lb. per square inch. 

Revolutions per minute =75. 

Determine dimensions necessary to prevent wear and heating and then 
test for strength and stiffness. 

5. Design a crank pin for a high-speed engine having the following 
dimensions and conditions: 

Diameter of piston . =14 in. 

Maximum steam pressure =100 lb. per square inch. 

Mean steam pressure =50 lb. per square inch. 

Revolutions per minute =250. 

6. Make a careful study and sketch of journals and journal bearings on 
each of the following machines and analyze as to (a) Materials, (b) Adjust- 
ment, (c) Lubrication. 

a. An engine lathe. 

b. A milling machine. 

c. A steam engine. 

d. An electric generator or motor. 

7. Sketch at least two forms of oil cup used in the laboratories and 
explain their working. 

8. The shaft journal of a vertical engine is 4 in. in diameter by 6 in. 
long. The cap is of cast iron, held down by 4 bolts of wrought iron, each 
5 in. from center of shaft, and the greatest vertical pressure is 12,000 lb. 

Calculate depth of cap at center for both strength and stiffness, and also 
the diameter of bolts. 

9. Investigate the strength of the cap and bolts of some pillow block 
whose dimensions are known, under a pressure of 500 lb. per square inch of 
projected area. 

10. The total weight on the drivers of a locomotive is 64,000 lb. The 
drivers are four in number, 5 ft. 2 in, in diameter, and have journals 7| in. 
in diameter. 

Determine horse-power consumed in friction when the locomotive is 
running 50 miles an hour, assuming tan W = .05. 

65. Step-Bearings. — Any bearing which is designed to resist 
end thrust of the shaft rather than lateral pressure is denomi- 



STEP BEARINGS 145 

nated a step or thrust bearing. These are naturally most used 
on vertical shafts, but may be frequently seen on horizontal 
ones as for example on the spindles of engine lathes, boring 
machines and milling machines. 

Step-bearings may be classified according to the shape of the 
rubbing surface, as flat pivots and collars, conical pivots, and 
conoidal pivots of which the Schiele pivot is the best known. 
When a step-bearing on a vertical shaft is exposed to great pres- 
sure or speed it is sometimes lubricated by an oil tube coming up 
from below to the center of the bearing and connecting with a 
stand pipe or force-pump. The oil entering at the center is 
distributed by centrifugal force. 

66. Friction of Pivots or Step -bearings. — Flat Pivots. 

Let W = weight on pivot 

c?i= outer diameter of pivot 
p = intensity of vertical pressure 
T = moment of friction 
/= coefficient of friction = ton (p. 

We will assume p to be a constant which is no doubt approxi- 
mately true. 

_ w m 

Then p = = —7^ 

area iia{ 

Let r = the radius of any elementary ring of a width = dr, 

then area of element = 2 Trrdr 
Friction of element =/pX 2 Trrdr 
Moment of friction of element = 2/2? 7rr^(ir 
and 



T = 



/a 
— r^dr (a) 





or r = 2/p;r^ = 2/^;rg 

24 ^ndf 3^-^ '' ^ ^ 

The great objection to this form of pivot is the uneven wear 
due to the difference in velocity between center and circum- 
ference. 



146 MACHINE DESIGN 

67. Flat Collar. 

Let ^2 = inside diameter 
Integrating as in equation (a) above, but using limits 

-^ and ^ we have 



T = 2/p7r 



d^-d 



24 



In this case 



P 



4:W 



and 



7r{dl-dl) 



(82) 



68. Conical Pivot. 

Let a= angle of inclination to the vertical. 
As in the case of a flat ring the intensity of the vertical pres- 
sure is 

4TF 

'^~7t{dl-dl) 

and the vertical pressure on an elementary ring of the bearing 
surface is 




W dP 



dW = 



Fig. 60. 
4W 



X27:rdr = 



SWrdr 



7t{dl-dl)'^ df-dl 

As seen in Fig. 60 the normal pressure on the elementary ring 



IS 



dP = 



dW 



SWrdr 



sina {dl—d2)sina 



STEP BEARINGS 147 

The friction on the ring isfdP and the moment of this friction is 



{dl—diysina 
d. 



T= ... ^^J . I r'dr 



I 



{dl—dl)sina I d 



_, Wf d\-d% 
^ sina dl—dl' 



(83) 



As a approaches ^ the value of T approaches that of a flat rmg, 

and as a approaches the value of T approaches oo . 
If ^2 = we have 



T = i^M' (84) 



sina 



The conical pivot also wears unevenly, usually assuming a 
concave shape as seen in profile. 

69. Schiele's Pivot. — By experimenting with a pivot and 
bearing made of some friable material, it was shown that the 
outline tended to become curved as shown ^ ^^ 

in Fig. 62. This led to a mathematical in- i (J 

vestigation which showed that the curve j ^ 

would be a tractrix under certain conditions. 

This curve may be traced mechanically as 
shown in Fig. 61. 

Let the weight W be free to move on a 
plane. Let the string SW be kept taut and '" p « 61 
the end S moved along the straight line SL. 
Then will a pencil attached to the center of W trace on the 
plane a tractrix whose axis is SL. 

In Fig. 62. let /STF = length of string = r ^ and let P be any point 
in the curve. Then it is evident that the tangent PQ to the curve 
is a constant and = r^ 

Also -^—p, = r.. 

sino 



148 



MACHINE DESIGN 



Let a pivot be generated by revolving the curve around its 
axis SL. As in the case of the conical pivot it can be proved 
that the normal pressure on an element of convex surface is 

ip_ SWrdr 

{dl-dl)sind' ^^^ 

Let the normal wear of the pivot be 
assumed to be proportional to this nor- 
mal pressure and to the velocity of the 
rubbing surfaces, i.e. normal wear pro- 
portional to pr, then is the vertical 




wear proportional to 



But 



IS 



Fig. 62. 



dP = 



dT = 



sind sind 

a constant, therefore the vertical wear 
will be the same at all points. This 
is the characteristic feature and advan- 
tage of this form of pivot. 
As shown in equation (a) 

SWr^dr 
df^J 
SWfVj^ rdr 



and 



T= 



dl —d 

8TF/ri 



Wfd, 



(85) 



dl-dl 2 2 

T is thus shown to be independent of c?2 o^ of the length of 
pivot used. 

This pivot is sometimes wrongly called antifriction. As will 
be seen by comparing equations (81) and (85) the moment of 
friction is 50 per cent greater than that of the common flat 
pivot. 

The distinct advantage of the Schiele pivot is in the fact 
that it maintains its shape as it wears and is self-adjusting. It 
is an expensive bearing to manufacture and is seldom used on 
that account. 

It is not suitable for a bearing where most of the pressure is 
sideways. 

Mr. H. G. Reist of the General Electric Company, in the paper 
before alluded to, explains the practice of that company in 
regard to large step bearings for steam turbine work. 



STEP BEARINGS 



149 



The pressures and speeds allowed are the same as already 
quoted for cylindrical bearings. The bearings are usually sub- 
merged in oil and are provided with radial grooves in the step 
j ournal to force the oil over the entire surface. 

Two bearings are sometimes employed, one above the other, 
one being supported by a spring so as to take about one-half 
the load. 

If pressure and speed are great, the weight is supported on a 
film of oil or water maintained by pressure. A circular recess 
about half the diameter of the bearing disc allows the oil to 
distribute. The distance that the bearing is raised by the oil 
pressure is from .003 to .005 in. and the pressures employed 
vary from 250 to 800 lb. per square inch according to circum- 
stances. The initial pressure to raise the step will be about 25 
per cent greater. The following figures are quoted as examples 
of ordinary practice. 



Weight of rotor 

Revolutions per minute 

Diameter of bearing 

Pressure of oil 

Quantity of oil in gallons per minute 



9,800 

1,800 

9.75 

180 

1 



53,000 

750 

16 

420 

3.5 



187,000 

500 

22.5 

650 

6 



70. Multiple Bearings. — To guard against abrasion in flat 
pivots a series of rubbing surfaces which divide the wear is some- 
times provided. Several flat discs placed beneath the pivot and 
turning indifferently may be used. Sometimes the discs are 
made alternately of a hard and a soft material. Bronze, steel 
and raw hide are the more common materials. 

Notice in this connection the button or washer at the outer end 
of the head spindle of an engine lathe and the loose collar on the 
main journal of a milling machine. See Figs. 54 and 55. Pivots 
are usually lubricated through a hole at the center of the bearing 
and it is desirable to have a pressure head on the oil to force it in. 

The hydraulic foot step sometimes used for the vertical shafts 
of turbines is in effect a rotating plunger supported by water 
pressure underneath and so packed in its bearing as to allow a 



150 



MACHINE DESIGN 



slight leakage of water for cooling and lubricating the bearing 
surfaces. 

The compound thrust bearing generally used for propeller 
shafts consists of a number of collars of the same size forged on 
the shafts at regular intervals and dividing the end thrust between 
them, thus reducing the intensity of pressure to a safe limit 
without making the collars unreasonably large. 

Fig. 63 shows the shape of the horseshoe rings for bearing 
surface arranged for independent water cooling. 




Fig. 63. 



A safe Talue for p the intensity of pressure is, according to 
Whitham, 60 lb. per square inch for high-speed engines. 

A table given by Prof. Jones in his book on Machine Design 
shows the practice at the Newport News ship-yards on marine 
engines of from 250 to 5000 horse-power. The outer diameter 
of collars is about one and one-half times the diameter of the 
shafts in each case and the number of collars used varies from 
6 in the smallest engine to 11 in the largest. The pressure per 
square inch of bearing surface varies from 18 to 46 lb. with an 
average value of about 32 lb. 

Mr. G. W. Dickie gives some data concerning modern naval 
practice in the design of thrust bearings. The usual method of 
determining the pressure is to assume two-thirds of the indi- 
cated horse-power and calculate the pressure from that by 
the formula : 



THRUST BEARINGS 



151 



P = l HP 



2X60X33000 
3X6080 aS 



(86) 



where P = pressure on thrust bearing >S = speed of ship in 
knots. 

Mr. Dickie quotes examples from modern practice for both 
naval and merchant service. These are assembled in Table 
XXXIX. 

All of these bearings except No. 4 were supplied with water cir- 
culation through each horseshoe. No. 3 required especial care 
when running on account of the high rubbing velocity. 

TABLE XXXIX 

Properties of Marine Thrust Bearings (Dickie) 



Data 


1 


2 


3 


4 


Armored 
cruiser 


Protected 
cruiser 


Torpedo boat 
destroyer 


Passenger 
steamer 


Speed, kaots 


22 

1,188 

11,500 

112,700 

95 

642 


22.5 

891 

6,800 

89,000 

100 

610 


28 

581 

4,200 

33,600 

58 
827 


21 

2,268 

15,000 

154,500 

68.1 

504 


Surface of ring (square inch) 

Horse-power, one engine 


Total thrust (pounds) 


Pressure per square inch (pounds) . 
Mean rubbing speed (feet per 
minute) . 



PROBLEMS 



1. Design and draw to full size a Schiele pivot for a water wheel shaft 
4 in. in diameter, the total length of the bearing being 3 in. 

Calculate the horse-power expended in friction if the total vertical 
pressure on the pivot is two tons and the wheel makes 150 revolutions per 
minute and assuming/ =.2 5 for metal on wet wood. 

2. Compare the friction of the pivot in Prob. 1, with that of a flat collar 
of the same projected area and also with that of a conical pivot having 
a — 30 degrees. 

3. Design a compound thrust bearing for a propeller shaft the diameters 
being 14 and 21 in., the total thrust being 80,000 lb. and the pressure 60 lb. 
per square inch. 

Calculate the horse-power consumed in friction and compare with that 
developed if a single collar of same area had been used. Assume/ =.05 
and revolutions per minute = 120. 



152 MACHINE DESIGN 



REFERENCES 



Machine Design. Low and Bevie, Chapter IX. 

Steam Engine. Ripper, Chapter XVII. 

Large Experimental Bearing. Am. Mach., March 15, 1906. 

Experiments on Bearings. Am. Mach., Oct. 18, 1906. 

Lubrication of Bearings. Tr. A. S. M. E., Vol. X, p. 810; Vol. XIII, 

p. 374. 
Lubrication. Eng. Mag., Dec, 1908. 
Bearings, a Symposium. Tr. A. S. M. E., Vol. XXVII, p. 420. 



CHAPTER VIII 
BALL AND ROLLER BEARINGS 

71. General Principles. — The object of interposing a ball or 
roller between a journal and its bearing, is to substitute rolling 
for sliding friction and thus to reduce the resistance. This can 
be done only partially and by the observance of certain principles. 
In the first place it must be remembered that each ball can roll 
about but one axis at a time; that axis must be determined and 
the points of contact located accordingly. 

Secondly, the pressure should be approximately normal to the 
surfaces at the points of contact. 

Finally it must be understood, that on account of the contact 
surfaces being so minute, a comparatively slight pressure will 
cause distortion of the balls and an entire change in the 
conditions. 




72. Journal Bearings. — These may be either two, three or 
four point, so named from 
the number of points of con- 
tact of each ball. 

The axis of the ball may 
be assumed as parallel or in- 
clined to the axis of the jour- 
nal and the points of contact 
arranged accordingly. The 
simplest form consists of a plain cylindrical journal running 
in a bearing of the same shape and having rings of balls inter- 
posed. The successive rings of balls should be separated 
by thin loose collars to keep them in place. These collars are 
a source of rubbing friction, and to do away with them the 
balls are sometimes run in grooves either in journal, bearing or 
both. 

Fig. 64 shows a bearing of this type, there being three points 
of contact and the axis of ball being parallel to that of journal. 

153 



154 



MACHINE DESIGN 



/— — v^ 



The bearings so far mentioned have no means of adjustment 
for wear. Conical bearings, or those in which the axes of the 
balls meet in a common point, supply this deficiency. In 
designing this class of bearings, either for side or end thrust, the 
inclination of the axis is assumed according to the obliquity 
desired and the points of contact are then so located that there 
shall be no slipping. 

Fig. 65 illustrates a common form of adjustable or cone bearing 
and shows the method of designing a three-point contact. A C 
is the axis of the cone, while the shaded area is a section of the 
cup, so called. Let a and h be two points of contact between 

ball and cup. Draw the line a h 
and produce to cut axis in A. 
Through the center of ball draw 
the line A B; then will this be 
the axis of rotation of the ball 
- and a c,h d will be the projections 
of two circles of rotation. As the 
radii of these circles have the 
same ratio as the radii of revolu- 
tion a n, h m, there will be no 
slipping and the ball will roll as a 
cone inside another cone. The 
exact location of the third point of contact is not material. If 
it were at c, too much pressure would come on the cup at &; if 
at d there would be an excess of pressure at a, but the rolling 
would be correct in either case. A convenient method is to 
locate p by drawing A D tangent to ball circle as shown. It is 
recommended, however, that the two opposing surfaces at p 
and 6 or a should make with each other an angle of not less than 
25 degrees to avoid sticking of the ball. 

To convert the bearing just shown to four-point contact, it 
would only be necessary to change the one cone into two cones 
tangent to the ball at c and d. 

To reduce it to two-point contact the points a and b are 
brought together to a point opposite p. As in this last case the 
ball would not be confined to a definite path it is customary to 
make one or both surfaces concave conoids with a radius about 
three-fourths the diameter of the ball. See Fig. 66. 




BALL BEARINGS 



155 



73. Step -bearings. — The same principles apply as in the pre- 
ceding article and the axis and points of contact may be varied 
in the same way. The most common form of step-bearing con- 
sists of two flat circular plates separated by one or more rings of 
balls. Each ring must be kept in place by one or more loose 
retaining collars, and these in turn are the cause of some sliding 




Fig. 66. 



Fig. 67. 



friction. This is a bearing with two-point contact and the balls 
turning on horizontal axes. If the space between the plates is 
filled with loose balls, as is sometimes done, the rubbing of the 
balls against each other will cause considerable friction. 

To guide the balls without rubbing friction three-point contact 
is generally used. 




Fig. 67 illustrates a bearing of this character. The method 
of design is shown in the figure, the principle being the same as 
in Fig. 65. By comparing the lettering of the two figures the 
similarity will be readily seen. 



156 



MACHINE DESIGN 



This last bearing may be converted to four-point contact by 
making the upper collar of the same shape as the lower. 

What is practically a two-point contact with some of the 
advantages of four point is attained by the use of curved races for 
the balls as in Fig. 68. 

To insure even distribution of the load, the lower ring is 

supported on a self-adjusting spher- 
ical collar. The radii of the curved 
races should not be less than two- 
thirds the diameter of the balls. 

To guide the balls in two-point 
contact use is sometimes made of a 
cage ring, a flat collar drilled with 
holes just a trifle larger than the 
balls and disposing them either in 
spirals or in irregular order. See 
Fig. 69. 

This method has the advantage of 
making each ball move in a path of different radius thus secur- 
ing more even wear for the plates. 




Fig. 69. 



74. Materials and Wear. — The balls themselves are always 
made of steel, hardened in oil, tempered and ground. They 
are usually accurate to within one ten-thousandth of an inch. 
The plates, rings and journals must be hardened and ground 
in the same way and perhaps are more likely to wear out 
or fail than the balls. A long series of experiments made at 
the Case School of Applied Science on the friction and 
endurance of ball step-bearings showed some interesting 
peculiarities. 

Using flat plates with one circle of quarter-inch balls it was 
found that the balls pressed outward on the retaining ring with 
such force as to cut and indent it seriously. This was probably 
due to the fact that the pressure slightly distorted the balls and 
changed each sphere into a partial cylinder at the touching points. 
While of this shape it would tend to roll in a straight line or a 
tangent to the circle. Grinding the plates slightly convex at an 
angle of 1 to 1^ degrees obviated the difficulty to a certain 
extent. Under even moderately heavy loads the continued 



- BALL BEARINGS 157 

rolling of the ring of balls in one path soon damaged the plates 
to such an extent as to ruin the bearing. 

A flat bearing filled with loose balls developed three or four 
times the friction of the single ring and a three-point bearing 
similar to that in Fig. 68 showed more than twice the friction of 
the two point bearing. 

A flat ring cage such as has already been described was the 
most satisfactory as regards friction and endurance. 

The general conclusions derived from the experiments were 
that under comparatively light pressures the balls are distorted 
sufficiently to disturb seriously the manner of rolling and that 
it is the elasticity and not the compressive strength of the balls 
which must be considered in designing bearings. 

75. Design of Bearings. — Figures on the direct crushing 
strength of steel balls have little value for the designer. For 
instance it has been proved by numerous tests that the average 
crushing strengths of ^-in. and |-in. balls are about 7500 lb. 
and 15^000 lb. respectively. Experiments made by the writer 
show that a J-in. ball loses all value as a transmission element 
on account of distortion, at any load of more than 100 lb. 

Prof. Gray states, as a conclusion from some experiments 
made by him, that not more than 40 lb. per ball should be allowed 
for f-in. balls. 

This distortion doubtless accounts for the failure of theoretic- 
ally correct bearings to behave as was expected of them. 

Mr. Charles R. Pratt reports the limit of work for i-in. balls 
in thrust bearings to be 100 lb. per ball at 700 revolutions per 
minute and 6 in. diameter circle of rotation. 

Mr. W. S. Rogers gives the maximum load for a I-Id. ball as 
1000 lb. and for a ^-in. ball as 200 lb. 

76. Endurance of Ball Bearings. — ^For complete and reliable 
data on the strength and endurance of ball bearings, reference 
is made to a paper by Mr. Henry Hess and to translations of the 
work of Professor Stribeck.^ 

' The formulas which follow are derived mainly from the sources 
mentioned. 

1 Trans. A. S. M. E., Vol. XXIX. 



158 



MACHINE DESIGN 



Ball bearings do not fail from wear but, as already noticed, 
from distortion and injury at the contact points. The use of a 
curved race, as in Fig. 68, will increase the durability, because 
the contact point is reinforced by the material at either side. 
In a journal bearing having one ring of balls, one-fifth of the 
total number of balls is considered as carrying the load. In a 
plain journal, the unit is the square inch of projected area. In 
a ball bearing, for projected area is substituted the square of 
ball diameter multiplied by one-fifth the number of balls. 
Let d = diameter of ball in inches 
n = number of balls in ring 
TF = total load on balls 
p = safe load on one ball. 

p = —-- = kd^ (a) 



Then 



n 



where k is sl constant depending on the material and the type of 
bearing. 
From equation (a) : ' 



W=k 



(87) 



nd^ 



where —^ corresponds to the (Id) or projected area of the plain 
o 

bearing. The values of k are as follows : 



Shape of race 


Hardened steel balls 


Hardened steel alloy balls 


Flat 


500 to 700 
1,500 


700 to 1,000 
2,000 


Curved to radius = f <i . 



The load capacity of balls may be affected by various condi- 
tions; lack of uniformity in the hardness of either balls or race 
will reduce the capacity; lack of uniformity in size of balls is 
also a source of inefficiency; sudden variations of speed cause 
shocks which impair the capacity of the bearing. 

The ball bearing is of somewhat the same nature as a chain or a 
gear and weakness of any unit leads to the destruction of the 
whole. 



ROLLER BEARINGS 



159 



The average coefficient of friction of a good ball bearing is 
about 0.0015. 

Speeds of over 1500 revolutions per minute are impracticable. 

77. Roller Bearings. — The principal disadvantage of ball 
bearings lies in the fact that contact is only at a point and that 
even moderate pressure causes excessive distortion and wear. 
The substitution of cylinders or cones for the balls is intended to 
overcome this difficulty. 

The simplest form of roller bearing consists of a plain cylindrical 
journal and bearing with small cylindrical rollers interposed 
instead of balls. There are two 



x\\^^V\^^^^V\^xV^^\\\\^^\\\^^^^^^^ 



21 



^\\\\\\V\\\\\\\\\\\\\\^^^^^^ 



Fig. 70. 



difficulties here to be overcome. 
The rollers tend to work end- i 
ways and rub or score whatever ^ 
retains them. .They also tend 
to twist around and become 
unevenly worn or even bent and 
broken, unless held in place by some sort of cage. In short they 
will not work properly unless guided and any form of guide en- 
tails sliding friction. The cage generally used is a cylindrical 
sleeve having longitudinal slots which hold the rollers loosely 
and prevent their getting out of place either sideways or endways. 
The use of balls or convex washers at the ends of the rollers 
has been tried with some degree of success. See Fig. 70. Large 

rollers have been turned smaller at the 
ends and the bearings then formed 
allowed to turn in holes bored in re- 
volving collars. These collars must 
be so fastened or geared together 
as to turn in unison. 

78. Grant Roller Bearing. — The 

Grant roller is conical and forms an 
intermediate between the ball and the cylindrical roller having 
some of the advantages of each. The principle is much the 
same as in the adjustable ball bearing, Fig. 65, rolling cones 
being substituted for balls, Fig. 71. The inner cone turns 
loose on the spindle. The conical rollers are held in position 




Fig. 71. 



160 MACHINE DESIGN 

by rings at each end, while* the outer or hollow cone ring is ad- 
justable along the axis. 

Two sets of cones are used on a bearing, one at each end to 
neutralize the end thrust, the same as with ball bearings. 

79. Hyatt Rollers. — The tendency of the rollers to get out of 
alignment has been already noticed. The Hyatt roller is in- 
tended by its flexibility to secure uniform pressure and wear 
under such conditions. It consists of a flat strip of steel wound 
spirally about a mandrel so as to form a continuous hollow 
cylinder. It is true in form and comparatively rigid against 
compression, but possesses sufficient flexibility to adapt itself to 
slight changes of bearing surface. 

Experiments made by the Franklin Institute show that the 
Hyatt roller possesses a great advantage in efficiency over the 
solid roller. 

Testing |-in. rollers between flat plates under loads increasing 
to 550 lb. per linear inch of roller developed coefficients of 
friction for the Hyatt roller from 23 to 51 per cent less than for 
the solid roller. Subsequent examination of the plates showed 
also a much more even distribution of pressure for the former. 

A series of tests were conducted by the writer in 1904-05 to 
determine the relative efficiency of roller bearings, as compared 
with plain cast-iron and Babbitted bearings under similar con- 
ditions.^ The bearings tested had diameters of 1||, 2^-^, 2^-^, 
and 2\^ in. and lengths approximately four times the diameters. 
In the first set of experiments Hyatt roller bearings were com- 
pared with plain cast-iron sleeves, at a uniform speed of 480 
revolutions per minute and under loads varying from 64 to 264 
lb. The cast-iron bearings were copiously oiled. 

As the load was gradually increased, the value of / the coeffi- 
cient of friction remained nearly constant with the plain bearings, 
but gradually decreased in the case of the roller bearings. Table 
XL gives a summary of this series of tests. 

The relatively high values of / in the 2 1\ and 2-J-f roller bearings 
were due to the snugness'of the fit between the journal and the 
bearing, and show the advisability of an easy fit as in ordinary 
bearings. 

» Mchy., N. Y., Oct., 1905. 



ROLLER BEARINGS 



161 



TABLE XL 

Coefficients of Friction for Roller and Plain Bearings 



Diameter of 
journal 


Hyatt bearing 


Plain bearing 


Max. 


! 
Min. Ave. 


Max. Min. 


Ave, 


m 

2^ 
2tV 
2H 


.036 
.052 
.041 
.053 


.019 
.034 
.025 
.049 


.026 
.040 
.030 
.051 


.160 
.129 
.143 
.138 


.099 
.071 
.076 
.091 


.117 
.094 
.104 
.104 



The same Hyatt bearings were used in the second set of 
experiments, but were compared with the McKeel solid roller 
bearings and with plain Babbitted bearings freely oiled. The 
McKeel bearings contained rolls turned from solid steel and 
guided by spherical ends fitting recesses in cage rings at each 
end. The cage rings were joined to each other by steel rods 
parallel to the rolls. The journals were run at a speed of 560 
revolutions per minute and under loads varying from 113 to 
456 lb. Table XLI gives a summary of the second series of 
tests. 

TABLE XLI 
Coefficients of Friction for Roller and Plain Bearings 



Diam. 

of 
journal 


Hyatt bearing 


McKeel bearing 


Babbitt bearing 


Max. 


Min. 


Ave. 


Max. 


Min. 


Ave. 


Max. 


Min. 


Ave. 


lA 
2A 

2tV 
211 


.032 
.019 
.042 
.029 


.012 
.011 
.025 
.022 


.018 
.014 
.032 
.025 


.033 


.017 


.022 


.074 
.088 
.114 
.125 


.029 
.078 
.083 
.089 


.043 
.082 
.096 
.107 


.028 
.039 


.015 
.019 


.021 

.027 



The variation in the values for the Babbitted bearing is due 
to the changes in the quantity and temperature of the oil. For 



162 



MACHINE DESIGN 



heavy pressures it is probable that the plain bearing might be 
more serviceable than the others. Notice the low values for / 
in Table XXXVII. 

Under a load of 470 lb. the Haytt bearing developed an end 
thrust of 13.5 lb. and the McKeel one of 11 lb. 

This is due to a slight skewing of the rolls and varies, some- 
times reversing in direction. 

If roller bearings are properly adjusted and not overloaded a 
saving of from two thirds to three-fourths of the friction may be 
reasonably expected. 

Professor A. L. Williston reports some tests of Hyatt roller 
bearings made at Pratt Institute in 1904. The journals were 
1.5 in. diameter and 4 in. long. The speeds varied from 128 to 
585 revolutions per minute. Both the roller and the plain 
bearings were lubricated with the same grade of oil. The total 
load on the bearing was gradually increased from 1900 lb. to 
8300 lb. The average friction of each bearing was as given in the 
table. 

TABLE XLII 
Coefficients of Friction for Roller and Plain Bearings 



Revolutions per ^t xx 
. ^ Hyatt 
minute 


Plain cast iron 


Plain bronze 


130 
302-320 
410-585 


.0114 
.0099 
.0147 


.0548 
.0592 
.0683 


.0576 
.0661 
.140 



In several instances the cast-iron bearing seized under pres- 
sures above 5000 lb. while the bronze bearing proved unreliable at 
pressures over 3000 lb. 

The roller bearing was further tested with total pressures 
from 10,800 lb. to 23,500 lb. at 215 revolutions per minute and 
the coefficient found to vary from .0094 to .0076. 



80. Roller Step -bearings. — In article 74 attention was called 
to the fact that the balls in a step-bearing under moderately 
heavy pressures tend to become cylinders or cones and to roll 



ROLLER BEARINGS 



163 



accordingly. This has suggested the use of small cones in place 
of the balls, rolling between plates one or both of which are also 
conical. A successful bearing of this kind with short cylinders 
in place of cones is used by the Sprague-Pratt Elevator Co., and 
is described in the American Machinist for June 27, 1901. The 
rollers are arranged in two spiral rows so as to distribute the 
wear evenly over the plates and are held loosely in a fiat ring 
cage. This bearing has run well 
in practice under loads double 
those allowable for ball bearings, 
or over 100 lb. per roll for rolls 
i in. in diameter and J in. long. 
Fig. 72 illustrates a bearing 
of this character. Collars simi- 
lar to this have been used in 
thrust bearings for propeller 
shafts. 




Fig. 72. 



81. Design of Roller Bearings. 

— Further reference is here made 

to the discussion mentioned in Art. 76 for information as to the 

design and construction of roller bearings. As in the case of 

ball bearings, one-fifth of all the rolls is assumed to carry the 

load and the area used for comparison may be expressed by the 

formula: 

7ild 
a = ^^ 
5 

where, 

n = number of rolls 

d = diameter of rolls in inches 
? = length of rolls in inches. 
The allowable pressure per roll is, 

p = kld (a) 

where k is b, constant depending on the material and shape of the 
roll. The whole load on the bearing is 

nld 



W = k- 



(88) 



Mr. Frank Mossberg gives the following values of the safe load 



164 



MACHINE DESIGN 



for roller bearings of the Mossberg type. These bearings have 
small solid steel rolls hardened to a spring temper and guided by 
bronze cages similar to those mentioned in Art. 77. The journal 
is tempered to a medium hardness and the box is of high carbon 
steel and very hard. 



TABLE XLIII 

Safe Load on Mossberg Roller Bearings 



Length of 

journal, 

inches 


Diameter 

of 

journal, 

inches 


Diameter 

of 

rolls, 

inches 


Number 

of 

rolls 


Safe load 

on 
journal, 
pounds 


Value of k 
bW 
nld 


I 




d 


n 


W 




3 

3.75 

4.5 

6 

7.5 

9 
10.5 
12 

13.5 
18 

22.5 
27 
30 
36 


2 

2.5 

3 

4 

5 

6 

7 

8 

9 
12 
15 
18 
20 
24 


1 

4 

.5 

1 6 
3 

8" 

t\ 

9 

Te- 
ll 

1 6 
13 
16 

1 
1 

H 

If 
If 

li 


20 
22 
22 
24 
24 
24 
22 
22 
24 
26 
28 
32 
34 
38 


3,500 

7,000 

13,000 

24,000 

37,000 

50,000 

70,000 

90,000 

115,000 

175,000 

255,000 

325,000 

400,000 

576,000 

Average 


1,170 
1,350 
1,750 
1,900 
1,830 
1,690 
1,860 
1,950 
1,770 
1,500 
1,470 
1,370 
1,300 
1,400 

1,590 



It will be noticed that k is not constant in Table XLIII, being 
greatest for the intermediate sizes. The average value is about 



A; = 1600. 



Smith and Marx give: 



k = 1000 for hardened steel 
A; =400 for cast iron. 



ROLLER BEARINGS 



165 



Mr. Mossberg considers one-third the entire number of rolls 
as bearing the load. This would make the formula read : 

nld 



W=k- 



(89) 



with an average value of A; = 960. 

The roller step-bearings of the same manufacture have small 
conical rolls with an angle of not over 6 or 7 degrees ; retaining rings 
or cages keep the rolls in correct positions. 

The bearing collars are of very hard high carbon steel and the 
rolls as in the journal bearing have a medium or spring temper. 
Table XLIV gives the proportions and safe loads. 

TABLE XLIV 
Safe Loads on Mossberg Roller Step-bearings 



Diameter 


Number 


Area of 


Safe load in 


pounds = W 


of shaft, 


of 


collar, 










inches, 
D 


rolls, 
n 


square 
inches 


75 revolutions 
per minute 


150 revolutions 
per minute 


2.25 


30 


10 


19,000 


9,500 


3.25 


30 


20 


40,000 


20,000 


4.25 


30 


35 


70,000 


35,000 


5.25 


30 


54 


108,000 


56,000 


6.50 


30 


78 


125,000 


62,000 


8.50 


32 


132 


200,000 


100,000 


9.50 


32 


162 


300,000 


150,000 



The formulas for pressure on rolls would be the same as in 
journal bearings except that the full number of rolls — n — would 
be effective at all times. 



W = knld 



(90) 



where I is the length of roll and d is its mean diameter. 

The table gives no information as to the proportions of the 
roll. 

The following proportions are scaled from a cut of the bearing: 



166 



MACHINE DESIGN 



Angle of cone about 7 degrees. 
l = O.SQD d = 0.1D 



Id^MQ D\ 



where 



D = diameter of shaft 
Z= working length of roll 
c? = mean diameter of roll. 

TABLE XLV 

Values of k for Roller Thrust Bearing 



D 


I 


d 


Id 


nld 


W 

Values of k =^rj 

nld 
















75 revolutions 


150 revolutions 












per minute 


per minute 


2.25 


0.81 


.225 


0.18 


5.46 


3,490 


1,745 


3.25 


1.17 


.325 


0.38 


11.4 


3,500 


1,750 


4.25 


1.53 


.425 


0.65 


19.5 


3,590 


1,795 


5.25 


1.89 


.525 


0.99 


29.7 


3,640 


1,820 


6.50 


2.34 


.650 


1.52 


45.6 


2,740 


1,370 


8.50 


8.06 


.850 


2.60 


83.3 


2,400 


1,200 


9.50 


3.42 


.950 


3.25 


104.0 


2,880 


1,440 



Space forbids reference to all of the many varieties of ball and 
roller bearings shown in manufacturers' catalogues. These are 
all subject to the laws and limitations mentioned in this chapter, 

While such bearings will be used more and more in the future, 
it must be understood that extremely high speeds or heavy 
pressures are unfavorable and in most cases prohibitive. 

Furthermore, unless a bearing of this character is carefully 
designed and well constructed it will prove to be worse than 
useless. 

REFERENCES 
Tests of Roller Bearings. Mchy., Oct., 1905. 

Tests of Ball Bearings. Am. Mach., Mar. 15, 1906, Jan. 23, 1908. 
Pressure on Balls. Am. Mach., July 12, 1906. 
Discussion on Ball Bearings. Power, July, 1907. 
Design of Roller Bearings. Power, Jan. 14, 1908. 
Recent Progress in Ball Bearings. Am. Mach., Nov. 7, 1907. 
Symposium. Tr. A. S. M. E., Vol. XXVII, p. 442. 
A Very Complete Discussion. Tr. A. S. M. E., Vol. XXIX, p. 367. 



CHAPTER IX 

SHAFTING, COUPLINGS AND HANGERS 

82. Strength of Shafting. 

Let D = diameter of the driving pulley or gear 
N = number revolutions per minute 
P=force applied at rim 
T' = twisting moment. 
The distance through which P acts in one minute is ttDN in. 
and wovk = PttDN in. lb. per minute. 

PD 

But —^ = T the moment, and 27rA^ = the angular velocity. 

.'. work = moment X angular velocity. 
One horse-power =33,000 ft. lb. per min. 
= 396,000 in. lb. per min. 
P7iDN__2nTN^ 
' ' 396000 "396000 

also r = ^^^ (92) 



DN 



(93) 



The general formula for a circular shaft exposed to torsion 
alone is 

But r=^^by(92) 

where N = no. rev. per min. 
Substituting in formula for d 



a^ij^^^H^ne^rly. (94) 

167 



168 



MACHINE DESIGN 



S may be given the following values: 

45,000 for common turned shafting. 

50,000 for cold rolled iron or soft steel. 

65,000 for machinery steel. 
It is customary to use factors of safety for shafting as follows : 

Headshafts or prime movers 15 

Line shafting 10 

Short counters 6 

The large factor of safety for head shafts is used not only on 
account of the severe service to which such shafts are exposed, 
but also on account of the inconvenience and expense attendant 
on failure of so important a part of the machinery. The factor of 
safety for line shafting is supposed to be large enough to allow 
for the transverse stresses produced by weight of pulleys, pull of 
belts, etc., since it is impracticable to calculate these accurately 
in most cases. 

Substituting the values of S and introducing factors of safety, 
we have the following formulas for the safe diameters of the 
various kinds of shafts. 

TABLE XL VI 
Diameters of Shafting 



Kind of shaft 


Material 


Common iron 


Soft steel 


Mach'y steel 


Head shaft 




4.00 ^'^Z 
3.38 i]^^ 


4.20 i]f 

3.67 f; 


Line shaft 


Counter shaft 





The Allis-Chalmers Co. base their tables for the horse power 
of wrought iron or mild steel shafting on the formula HP = cd^N 
where c has the following values: 



SHAFTING ' 169 

c 

Heavy or main shafting 008 

Shaft carrying gears 010 

Light shafting with pulleys , . 013 

This is equivalent to using values of S as 2570 lb., 3200 lb. 
and 4170 lb. per square inch in the respective classes — and would 
give for coefficients in Table XL VI -the numbers 5, 4.64 and 
4.25 which are somewhat larger than those given for similar cases 
in the table. 

A table published by Wm. Sellers & Co. in their shafting 
catalogue — gives the horse-powers of iron and steel shafts for 
given diameters and speeds. An investigation of the table 
shows it to be based upon a value of about 4000 lb. for aS or a 
coefficient of 4.31 in Table XLVI. 

83. Combined Torsion and Bending. — It frequently happens 
that a shaft is subjected to bending as well as torsion; a familiar 
example of this is the case of an engine shaft which carries the 
twisting moment due to the crank effort and also bending mo- 
ments caused by the overhang of the crank and the weight of the 
fly-wheel. 

The direct stress due to the twisting is shear in the plane of 
the cross-section; the stresses due to the bending are primarily 
tension and compression parallel to the axis and at right angles 
to the shearing stress. The combination of these produces obli- 
que stresses varying in direction and intensity as the shaft 
revolves. It is desirable to find the maximum values of these 
oblique stresses whether shearing or tensile. 

Let p = direct stress due to bending 
5 = direct stress due to twisting 
>Sf = resultant tensile stress 
aSs = resultant shearing stress. 

Then is it shown in treatises on the mechanics of materials 
that the maximurti values of the resultant stresses are as follows:^ 

^^ = | + 2\/4gM^ (a) 

^ Merriman's Mechanics of Materials, p. 151. Slocum and Hancock's 
Strength of Materials, p. 116. 



170 MACHINE DESIGN 



2 

Let M = bending moment on shaft 
T = twisting moment on shaft. 
Then by formulas (5) and (8) p. 3, 

10.2M 



Ss=±^\/4q'-\-p' (b) 





^ d' 








5.1T 






Substituting these values in (a) and (b) and reducing, we have: 

K 1 / \ 




5 1 




(c) 




Ss^±-^\/T' + M\ 




(d) 


But the bending 


moment which would produce 

Std^ 
^^' 10.2 


a stress = 


= Stia: 


and the twisting 


moment which would produce 
7, _Ssd' 


a stress^ 


= 83 is: 



^' 5.1 

Combining these equations with (c) and (d) respectively and 
reducing: 

M, = i{M±-\/W+IP). (95) 

T,=\/T' + M\ (96) 

The method of designing a shaft subjected to both bending and 
twisting moments may thus be stated : Determine the diameter 
of shaft necessary to withstand safely a bending moment M^, 
Equation (95) ; also, calculate the diameter to safely resist a 
twisting moment T^ (Equation (96)). The larger diameter 
would then be used, i.e., 

Equations (a) and (b) in this article may be used in combining 
shearing and tensile or shearing and compressive stresses, in 
whatever manner produced. 



COUPLINGS 



171 



Other examples of combined stresses are furnished by columns 
and by machine frames having eccentric loads (see Art. 17). 

In the case of columns, where the load is assumed to be central, 
the empirical formulas (12) and (12-a) given on pp. 4 and 5 
are recommended. 

Where a material like cast iron is concerned, as in the case of 
machine frames, no theoretical analysis is of much value and 
reliance can be placed only on experimental determinations of 
stresses and breaking loads. 



84. Couplings. — The flange or plate coupling is most commonly 
used for fastening together adjacent lengths of shafting. 

Fig. 73 shows the proportions 
of such a coupling. The flanges 
are turned accurately on all 
sides, are keyed to the shafts 
and the two are centered by the 
projection of the shaft from one 
part into the other as shown at 
A. The bolts are turned to fit 
the holes loosely so as not to 
interfere with the alignment. 

The projecting rim as at 5 pre- 
vents danger from belts catching on the heads and nuts of the 
bolts. 

The faces of this coupling should be trued up in a lathe after 
being keyed to the shaft. 

Jones and Laughlins in their shafting catalogue give the 
following proportions for flange couplings. 




Diam. of shaft 


Diam. of hub 


Length of hub 


Diam. of 
coupUng 


2 


^ 


3i 


8 


2i 


5f 


4f 


10 


3 


6f 


5i 


12 


3i 


8 


6i 


14 


4 


9 


7 


16 


5 


Hi 


81 


20 



172 



MACHINE DESIGN 



There are five bolts in each coupling. 

The sleeve coupling is neater in appearance than the flange 
coupling but is more complicated and expensive. . 

In Fig. 74 is illustrated a neat and effective coupling of this 
type. It consists of the sleeve S bored with two tapers and two 
threaded ends as shown. The. two conical, split bushings BB 




are prevented from turning by the feather key K and are forced 
into the conical recesses by the two threaded collars CC and 
thereby clamped firmly to the shaft. The key K also nicks 
slightly the center of the main sleeve S, thus locking the whole 
combination. 

Couplings similar to this have been in use in the Union Steel 
Screw Works, Cleveland, Ohio, for many years and have given 
good satisfaction. 

The Sellers coupling is of the 
type illustrated in Fig. 74, but is 
tightened by three bolts running 
parallel to the shaft and taking the 
place of the collars CC. 

In another form of sleeve coup- 
ling the sleeve is split and clamped 
to the shaft by bolts passing 
through the two halves as illus- 
trated in Fig. 75. 

The ''muff" coupling, as its name implies is a plain sleeve 
slipped over the shafts at the point of junction, accurately 
fitted and held by a key running from end to end. It may be 
regarded as a permanent coupling since it is not readily removed. 



t 





"1 '~ 





1 

D 


rrti 


- - 


rMi 




1 










M4J 


.J L. 


LUJ 





Fig. 75. 



CLUTCHES 



173 



85. Clutches. — By the term clutch, is meant a coupling which 
may be readily disengaged so as to stop the follower shaft or 
pulley. Clutch couplings are of two kinds, positive or jaw 
clutches and friction clutches. 

The jaw clutch consists of two hubs having sector shaped 
projections on the adjacent faces which may interlock. One 
of the couplings can be slid on its shaft to and from the other by 
means of a loose collar and yoke, so as to engage or disengage 
with its mate. This clutch has the serious disadvantage of not 
being readily engaged when either shaft is in motion. Friction 
clutches are not so positive in action, but can be engaged without 
difficulty and without stopping the driver. 

Three different classes of friction clutches may be distinguished 
according as the engaging 
members are flat rings, cones 
or cylinders. 

The Weston clutch. Fig. 
76, belongs to the first-named 
class. A series of rings inside 
a sleeve on the follower B in- 
terlocks with a similar series 
outside a smaller sleeve on 
the driver A somewhat as in 
a thrust bearing (Art. 70). 
Each ring can slide on its sleeve but must rotate with it. 

When the parts A and B are forced together the rings close up 
and engage by pairs, producing a considerable turning moment 
with a moderate end pressure. Let: 

P = pressure along axis 
71 = number of pairs of surfaces in contact 
/= coefficient of friction 
r = mean radius of ring 
7" = turning moment 

Then will: 

T = Pfnr. (97) 

If the rings are alternately wood and iron, as is usually the case, 
/ will have values ranging from 0.25 to 0.50. 

The cone clutch consists of two conical frustra, one external 




Fig. 76. 



174 



MACHINE DESIGN 



and one internal, engaging one another and driving by friction. 
Using the same notation as before, and letting a = angle between 
element of cone and axis, the normal pressure between the two 

p p/ 



surfaces will be: 



Therefore : 



8in a 



and the friction will be : 



sin a 



T 



Pfr 



sin a 



(98) 




^^^^^^^ 



a should slightly exceed 5 degrees to prevent sticking and /will 
be at least 0.10 for dry iron on iron. 

Substituting /= 0.10 and sin a =0.125 we have 2^ = 0.8 Pr as a 
convenient rule in designing. 

Fig. 77 illustrates the type of clutch more generally used on 
shafting for transmitting moderate quantities of power. 

As shown in the figure one 
member is attached to a loose 
pulley on the shaft, but this 
same type can be used for 
connecting two independent 
shafts. 

The ring or hoop H, 
finished inside and out, is 
gripped at intervals by pairs 
of jaws JJ having wooden 
faces. 

These jaws are actuated 
as shown by toggles and 
levers connected with the 
slip ring R. The toggles are 
so adjusted as to pass by the 
center and lock in the gripping position. 

These clutches are convenient and durable but occupy con- 
siderable room in proportion to their transmitting power. The 
Weston clutch is preferable for heavy loads. 

Cork inserts in metal surfaces have been used to some extent, 
as the coefficient of friction is much greater for cork than for 
wood. The cork may be boiled to soften it and forced into holes 
in one of the members. When pressure is applied, the projecting 
cork takes the load and carries it with good efficiency. As the 




Fig. 77. 



CLUTCHES 



175 



normal pressure is increased, the cork yields, finally becoming 
flush with the metal surface and dividing its load with the latter. 

Cork in its natural state is liable to wear quite rapidly under 
hard service. It may be hardened by being heated under heavy 
pressure and in this condition is much more durable. 

Professor I. N. Hollis gives the coefficients of friction for dif- 
ferent materials used in clutches, as follows : 

Cast iron on cast iron 0. 16 

Bronze on cast iron 0.14 

Cork on cast iron . 33 

Professor C. M. Allen in experiments on clutches for looms 
found that cork inserts gave a torque nearly double that of a 
leather face on iron. 

The roller clutch is much used on automatic machinery as it 
combines the advantages of positive driving and friction engage- 
ment. A cylinder on the follower is embraced by a rotating 
ring carried by the driver. 

The ring has a number of recesses cfn its inner surface which 
hold hardened steel rollers. These recesses being deeper at one 
end allow the rollers to turn freely as long as they remain in the 
deep portions. 

The bottom of the recess is inclined to the tangent of the circle 
at an angle of from 9 to 14 degrees. 

When by suitable mechanism the rollers are shifted to the 
shallow portions of the recesses they are immediately gripped 
between the ring and the cylinder and set the latter in motion. 

A clutch of this type is almost instantaneous in its action and 
is very powerful, being limited only by the strength of the 
materials of which it is composed. 

Several small rolls of different materials and diameters were 
tested by the writer in 1905 with the following results: 



Material 


Diameter 


Length 


Set load 


Ultimate load 


Cast iron 

Cast iron 

Cast iron 

Cast iron 

Soft steel 


0.375 

0.75 

1.125 

0.4375 

0.4375 

0.4375 


1.5 
1.5 
1.5 
1.5 
1.5 
1.5 


5,500 
6,800 
7,800 
8,800 
11,100 
35,000 


12,40.0 
19,500 
29,700 
20,000 


Hard steel 



176 MACHINE DESIGN 

86. Automobile Clutches. — The development of the automobile 
industry has created a demand for clutches of small size and 
considerable power; these clutches must also be capable of picking 
up a load gently and of holding it firmly; they must be durable 
and reliable under peculiarly severe conditions and for consider- 
able periods. 

Mr. Henry Souther contributes to the literature of this subject 
an interesting paper from which some of the following data are 
quoted. Reference is made to the paper itself for more complete 
information.^ 

Automobile clutches may be roughly classified as (a) conical; 
(b) disc or multiple disc; (c) band either expanding or contracting. 
The clutch is located between the engine and gear box, usually 
near the fly-wheel and sometimes forming a part of it. 

Conical clutches are in some respects the most satisfactory for 
automobile use. They require but slight motion for engagement 
and slight pressure to hold them in place. No lubricant is 
necessary and therefore there is no trouble from gumming and 
sticking. 

The materials used for the rubbing surfaces are generally 
aluminum covered with leather for one, and gray cast iron for the 
other. Castor or neatsfoot oil may be used to keep the leather 
soft. To render the engagement more gradual, springs are 
sometimes placed under the leather at six or eight points on the 
circumference; these permit some slipping until the whole 
surface of the leather is brought into contact. 

The angle of the cone is about 8 degrees in ordinary practice, 
but some manufacturers are using 10 or 12 degrees. (This is 
the angle on one side.) 

The principal difficulty with conical clutches is that of poor 
alignment. Unless the axes of the two cones coincide, engage- 
ment is uncertain and irregular. This coincidence can only be 
secured by the use of two universal joints insuring perfect 
flexibility. 

Mr. Souther gives the following table as representing three 
typical clutches in successful use: 

» Trans. A. S. M. E., May, 1908. 



CLUTCHES 



177 



TABLE XL VII 

Power of Clutches 





1 


2 


3 


Area of surface (square inches) 

Angle (one side) (degrees) 


113.1 

8 


78.7 
8 


73.6 
8 
7f 
250 
40 


Maximuni radius (inches) 


8h 8* 


Spring pressure (pounds) 

Horse-power 


375 

48 


320 

42 





Fig. 78 illustrates the conical clutch in its simplest form. 

The disc clutch consists of a disc on the driven member clamped 
between two discs on the driver, which latter is generally the 
fly-wheel. Springs are used to insure separation when dis- 
engaged and other springs furnish the pressure for engagement. 

A multiple disc clutch similar to the Weston is also used. In 
this case the discs are alter- 
nately of bronze and steel. 
All disc clutches must be 
lubricated and upon the 
type and quantity of lubri- 
cation depends the character 
of the service. Copious 
lubrication means gradual 
engagement and slight driv- 
ing power; scanty lubrica- 
tion gives more power and 
quick seizure. 

The principal disadvan- 
tage of the disc clutch is 
the heavy spring pressure 
necessary to insure driving 
power. 

The multiple disc clutches cause some trouble in lubrication 
and are complicated and difficult of access. 

Band clutches depend for their driving power on the friction 
between the case and an adjustable band or ring which can be 
expanded or contracted by suitable mechanism. 




Fig. 78. 



178 MACHINE DESIGN 

The more usual construction has a band which is expanded 
against the inside surface of the enclosing case by means of 
internally operated levers and springs. 

Centrifugal force at high speeds has a disturbing effect on 
the levers and sometimes causes the clutch to release auto- 
matically. This difficulty has been overcome in some clutches 
by an improved arrangement of levers and springs. 

87. Coupling Bolts. — The bolts used in the ordinary flange 
couplings are exposed to shearing, and the combined moment of 
the shearing forces should equal the twisting moment on the 
shaft. 

Let n = number of bolts 
d^ = diameter of bolt 
D = diameter of bolt circle. 

We will assume that the bolt has the same shearing strength 
as the shaft. The combined shearing strength of the bolts is 
.7854dJn>S and their moment of resistance to shearing is 

.7S54.dfnSx^ = . 3927 Dd^nS 

This last should equal the torsion moment of the shaft or 

&d^ 



.S927DdlnS = 



5.1 



Solving for d^ and assuming D = dd as an average value, we 

d 
have dj^ = ^=' (79) 

In practice rather larger values are used than would be given 
by the formula. 

88. Shafting Keys. — The moment of the shearing stress on a 
key must also equal the twisting moment of the shaft. 

Let b = breadth of a key 
Z = length of key 
/i = total depth of key 
iS' = shearing strength of key. 



SHAFTING KEYS 



179 



The moment of shearing stress on key is 



, , „, d hdlS' 



and this must equal 



5.1 



Usually h = 



d 



For shafts of machine steel S = S', and for iron shafts 5 = JaS' 
nearly, as keys should always be of steel. 
Substituting these values and reducing: 
For iron shafting Z = l. 2d nearly. 

For steel shafting l^l.Qd nearly as the least lengths 

of key to prevent its failing by shear. 

If the keyway is to be designed for uniform strength, the shear- 
ing area of the shaft on the line AB, Fig. 79, should equla the 
shearing area of the key, if shaft and key 
are of the same material and AB = CD — h. 

These proportions will make the depth 
of keyway in shaft about =^b and would 
be appropriate for a square key. 

To avoid such a depth of keyway which 
might weaken the shaft, it is better to use 
keys longer than required by preceding for- 
mulas. In American practice the total 
depth of key rarely exceeds f6 and one-half 
of this depth is in shaft. 

To prevent crushing of the key the moment of the compressive 
strength of half the depth of key must equal T. 




or 



d Ih _Sd^ 
2^2^^^" 5.1 



(a) 



where Sc is the compressive strength of the key. 

For iron shafts Sc^2S 

and for steel shafts 



^c — 2^ 



Substituting values of Sc and assuming h = %h = ^-^d we have 
Iron shafts Z = 2. 5(i nearly. 

Steel shafts l = '^\d nearly, as the least length for 

flat keys to prevent lateral crushing. 



180 



MACHINE DESIGN 





The above refers to parallel keys. Taper keys have parallel 
sides, but taper slightly between top and bottom. When 
driven home they have a tendency to tip the wheel or coupling 
on the shaft. This may be partially obviated by using two keys 
90 degrees apart so as to give three points of contact between 

hub and shaft. The taper of the keys 
is usually about { in. to 1 ft. 

The Woodruff key is sometimes 
used on shafting. As may be seen in 
Fig. 80 this key is semi-circular in 
shape and fits a recess sunk in the 
shaft by a milling cutter. 

89. Strength of Keyed Shafts. — Some 
very interesting experiments on the 
strength of shafts with keyways are 
reported by Professor H. F. Moore. ^ 
The material of the shafts was soft 
steel some being turned and some 
cold-rolled. The diameters varied from 
IJ to 21 in. Keyways of ordinary pro- 
portions, both for straight keys and for Woodruff keys, were cut 
in the specimens and the latter were then subjected to twisting 
and to combined twisting and bending. 

So far as the ultimate strength was concerned, the keyways 
seemed to have little effect, the shaft with a single keyway 
having about the same strength as a shaft without the keyway. 
After the elastic limit was passed, the keyways gradually closed 
up and were entirely closed at rupture. The elastic limit, 
however, was noticeably affected by the presence of a keyway. 
The ratio of the strength at elastic limit with keyway to the 
strength at elastic limit without keyway is called the efficiency 
and is denoted by -e-. The corresponding ratio of angles of 
twist inside the elastic limit is denominated -k-. 

According to Professor Moore, the following equations repre- 
sent fairly well the values of e and k : 



Fig. 80. 



e = l-0.2w-l.lh 

k = l+0Aw + 0.7h 

* University of Illinois Bulletin No. 42, 1909. 



(99) 
(100) 



SHAFTING KEYS 



181 



where w = 



width of keyway 



and h = 



diameter of shaft 
depth of keyway 



diameter of shaft 
Two values of w were used in the experiments: w = 0.25 and 0.50 
and two values of h: 

h = 0.125 and 0.1875. 

Table XLVIII gives the values of e as obtained by the experi- 
ments : 

TABLE XLVIII 

Efficiency of Shafts wifh Keyways 

„ . _ elastic strength of shaft with keyway 

elastic strength of shaft without keyway 



Dimensions of keyway 


W = 0.50 
/i = 0.125 


Tf = 0.25 
A = 0.1875 


1^^ = 0.25 
A = 0.125 


Woodruff 

System! 


Under simple torsion: 
Cold-rolled shaft, diameter, 

liin. 
Cold-rolled shaft, diameter, 

1 9/16 in. 


0.762 

0.803 
0.758 


0.760 

0.846 
0.817 


0.820 

0.900 
0.889 


0.840 

0.860 
0.815 


Cold-rolled shaft, diameter, 
1 15/16 in. 


0.748 
0.764 


0.710 
0.750 


0.860 
0.824 


0.826 
0.835 


Cold-rolled shaft, diameter, 
2iin. 


0.848 
0.705 


0.775 
0.689 


0.839 
0.825 


0.943 
0.861 


Under combined torsion and 
bending: 

1. Twisting moment = bend- 
ing moment. 

Cold-rolled shaft, diameter, 
li in. 


0.630 
0.680 


0.636 
0.698 


0.791 
0.803 


0.716 
0.750 


Cold-rolled shaft, diameter, 
1 15/16 in. 


0.584 
0.671 


0.697 
0.775 


0.854 


0.858 
0.840 




2, Twisting moment = 5/3 

bending moment. 
Cold-rolled shaft, diameter, 

li in. 


0.895 
0.870 


0.670 
0.735 


0.940 

0.888 


0.930 
0.880 


Cold-rolled shaft, diameter, 
1 15/16 in. 


0.740 
0.815 




0.832 
0.840 


0.856 
0.810 






General average 


0.752 


0.735 


0.850 


0.845 



^In 1 1/4-in. shafts keyways were cut for No. 15 Woodruff keys. 
In 1 9/16-in. shafts keyways were cut for No. 25 Woodruff keys. 
In 1 15/16-in. shafts keyways were cut for No. S Woodruff keys. 
In 2 1/4-in. shafts keyways were cut for No. U Woodruff keys. 



182 



MACHINE DESIGN 



The average value of the fiber stress of the cold-rolled shafting 
at the elastic limit was 38940 lb. and the average modulus of 
elasticity 11,985,000. 

It would appear that, considering the factor of safety usually 
allowed in shafting, the effect of ordinary keyways can safely be 
neglected. 

90. Hangers and Boxes. — Since shafting is usually hung to the 
ceiling and walls of buildings it is necessary to provide means 




Fig. 83. 



for adjusting and aligning the bearings as the movement of the 
building disturbs them. Furthermore as line shafting is contin- 
uous and is not perfectly true and straight, the bearings should 
be to a certain extent self-adjusting. Reliable experiments 



HANGERS 



183 




have shown that usuall}^ one-half of the power developed by an 
engine is lost in the friction of shafting and belts. It is important 
that this loss be prevented as far as possible. 

The boxes are in two parts and may be of bored cast-iron or 
lined with Babbitt metal. They are usually about four diam- 
eters of the shaft in length and are oiled by means of a well and 
rings or wicks. (See Art. 58.) 



The best method of support- 
ing the box in the hanger is 
by the ball-and-socket joint; 
all other contrivances such as 
set screws are but poor sub- 
stitutes. 
Fig. 81 shows the usual arrangement of the ball and socket. 
A and B are the two parts of the box. The center is cast in 
the shape of a partial sphere with C as a center as shown by the 
dotted lines. The two sockets S S can be adjusted vertically in 
the hanger by means of screws and lock nuts. The horizontal 

adjustment of the hang- 
er is usually effected by 
moving it bodily on the 
support, the bolt holes 
being slotted for this 
purpose. 

Counter shafts are 
short and light and are 
not subject to much 
bending. Consequently 
there is not the same 
need of adjustment as in 
^^^' ^6- line shafting. 

In Fig. 82 is illustrated a simple bearing for counters. The 
solid cast-iron box B with a spherical center is fitted directly in a 
socket in the hanger H and held in position by the cap C and a set 
screw. There is not space here to show all the various forms 
of hangers and floor stands and reference is made to the catalogues 
of manufacturers. Hangers should be symmetrical, i.e., the 
center of the^box should be in a vertical line with center of base. 
They should have relatively broad bases and should have the 




184 MACHINE DESIGN 

metal disposed to secure the greatest rigidity possible. Cored 
sections are to be preferred, 

Fig. 83 illustrates the proportions of a Sellers line-shaft hanger. 
This type is also made with the lower half removable so as to 
facilitate taking down the shaft. 

Fig. 84 shows the outlines of a hanger for heavy shafting as 
manufactured by the Jones & Laughlins Company while Fig. 85 
illustrates the design of the box with oil wells and rings. 

The open side hanger is sometimes adopted on account of the 
ease with which the shaft can be removed, but it is much less 
rigid than the closed hanger and is suitable only for light shafting. 
The countershaft hanger shown in Fig. 86 is simple, strong and 
symmetrical and is a great improvement over those using pointed 
set screws for pivots. Hangers similar to this are used by the 
Brown & Sharpe Mfg. Co. with some of their machines. 

PROBLEMS 

1. Calculate the safe diameters of head shaft and three line shafts for a 
factory, the material to be rolled iron and the speeds and horse-powers as 
follows : 

Head shaft 
Machine shop 
Pattern shop 
Forge shop 

2. Determine the horse-power of at least two lines of shafting whose 
speeds and diameters are known. 

3. Design and sketch to scale a flange coupling for a 3-in. line shaft 
including bolts and keys. 

4. Design a sleeve coupling for the foregoing, different in principle from 
the ones shown in the text. 

5. A 4-in. steel head shaft makes 100 rev. per min. Find the horse-power 
which it will safely transmit, and design a Weston ring clutch capable of 
carrying the load. 

There are to be six wooden rings and five iron rings of 12-in. mean diame- 
ter. Find the moment carried by each pair of surfaces in contact and the 
end pressure required. 

6. Find mean diameter of a single cone clutch for same shaft with same 
end pressure. 

7. Find radial pressure required for a clutch like that shown in Fig. 77, the 
ring being 24 in. in mean diameter and there being four pairs of grips. Other 
conditions as in preceding problems. 



100 H. P. 


200 rev. 


per min 


30 H. P. 


120 rev. 


per min 


50 H. P. 


250 rev. 


per min 


20 H. P. 


200 rev. 


per min 



HANGERS 185 

8. Select the line-shaft hanger which you prefer among those in the 
laboratories and make sketch and description of the same. 

9. Do. for a countershaft hanger. 

10. Explain in what way a floor-stand differs from a hanger. 

REFERENCES 

Machine Design. Low and Bevis, Chapter VIII. 

Efficiency of Shafting. Tr. A. S. M. E., Vol. VI, p. 461; Vol. VII, p. 138; 

Vol. XVIII, p. 228; Vol. XVIII, p. 861. 
Shafting Clutches. Tr. A. S. M. E., Vol. XIII, p. 236. 
Ball Bearing Hangers. Tr. A. S. M. E., Vol. XXXII, p. 533. 
Test of Clutch CoupHng. Tr. A. S. M. E., Vol. XXXII, p. 549. 



CHAPTER X 

GEARS, PULLEYS AND CRANKS 

91. Gear Teeth. — The teeth of gears may be either cast or cut, 
but the latter method prevails, since cut gears are more accurate 
and run more smoothly and quietly. The proportions of the 
teeth are essentially the same for the two classes, save that more 
back lash must be allowed for the cast teeth. The circular 
pitch is obtained by dividing the circumference of the pitch 
circle by the number of teeth. The diametral pitch is obtained 
by dividing the number of teeth by the diameter of the pitch 
circle and equals the number of teeth per inch of diameter. 
The reciprocal of the diametral pitch is sometimes called the 
module. The addendum is the radial projection of the tooth 
beyond the pitch circle, the dedendum the corresponding 
distance inside the pitch circle. The clearance is the difference 
between the dedendum and addendum; the back lash the differ- 
ence between the widths of space and tooth on the pitch circle. 

Let circular pitch =p 

module =-=m 

diametral pitch =-= — 
p m 

addendum =a 

dedendum or flank =/ 

clearance =f—a = c 

height —a-\-f=h 

width =w. (See Fig. 88.) 

The usual rule for standard cut teeth is to make w^^, allowmg 
no calculable back-lash, to make a = m and f=-^ oi" h = 2lm 
and clearance =^- 

o 
There is, however, a marked tendency at the present time 
toward the use of shorter teeth. The reasons urged for their 

186 



GEAR TEETH 



187 



adoption are: first, greater strength and less obliquity of action; 
second, less expense in cutting.^ Several systems have been 
proposed in which the height of tooth h varies from 0.4252? to 
0.552?. 

According to the latter system a = 0.25p, /^O.S/?, and c = .052). 

In modern practice the diametral pitch is a whole number or a 
common fraction and is used in describing the gear. For 
instance, a 3-pitch gear is one having 3 teeth per inch of diameter. 
The following table gives the pitches in common use and the 
proportions of long and short teeth. 

If the gears are cut, 'w = ^'j if cast gears are used, w = OA^'p 

to 0.48p. 

TABLE XLIX 

Proportions of Gear Teeth 



Pitch 


Standard teeth 


Short teeth 


Diametral 


Circular 


Addend, a 


Height h 


Clear- 
ance c 


Addend, a 


Height h 


Clear- 
ance c 


2 

4 

1 

n 
If 

2 

2\ 

2i 

2f 

3 

3i 

4 

5 

6 

7 

8 

9 
10 
11 
12 
13 
14 
15 
16 


6.283 
4.189 
3.142 
2.513 
2.094 
1.795 
1.571 
1.396 
1.257 
1.142 
1.047 
0.898 
0.785 
0.628 
0.524 
0.449 
0.393 
0.349 
0.314 
0.286 
0.262 
0.242 
0.224 
0.209 
0.196 


2. 

1.33 

1. 

0.8 

0.667 

0.571 

0.5 

0.445 

0.4 

0.364 

0.333 

0.286 

0.25 

0.2 

0.167 

0.143 

0.125 

0.111 

0.1 

0.091 

0.0834 

0.077 

0.0715 

0.0667 

0.0625 


4.25 

2.82 

2.125 

1.7 

1.415 

1.212 

1.062 

0.945 

0.85 

0.775 

0.708 

0.608 

0.531 

0.425 

0.354 

0.304 

0.266 

0.236 

0.212 

0.193 

0.177 

0.164 

0.152 

0.142 

0.133 


0.25 

0.167 

0.125 

0.1 

0.083 

0.071 

0.062 

0.056 

0.05 

0.045 

0.042 

0.036 

0.031 

0.025 

0.021 

0.018 

0.016 

0.014 

0.012 

0.011 

0.010 

0.010 

0.009 

0.008 

0.008 


1.571 
1.047 
0.785 
0.628 
0.524 
0.449 
0.392 
0.349 
0.314 
0.286 
0.262 
0.224 
0.196 
0.157 
0.131 
0.112 
0.098 
0.087 
0.079 
0.071 
0.065 
0.060 
0.056 
0.052 
0.049 


3.456 
2.303 
1.728 
1.383 
1.152 
0.988 
0.863 
0.768 
0.691 
0.629 
0.576 
0.494 
0.432 
0.345 
0.288 
0.246 
0.216 
0.191 
0.174 
0.156 
0.143 
• 0.132 
0.123 
0.114 
0.108 


0.314 

0.209 

0.157 

0.125 

0.105 

0.09 

0.078 

0.070 

0.063 

0.057 

0.052 

0.045 

0.039 

0.031 

0.026 

0.022 

0.020 

0.017 

0.016 

0.014 

0.013 

0.012 

0.011 

0.010 

0.010 



^ See Am. Mach. Jan. 7, 1897, p. 6. 



188 



MACHINE DESIGN 



92. Strength of Teeth. — Let P = total driving pressure on 
wheel at pitch circle. This may be distributed over two or more 
teeth, but the chances are against an even distribution. 

Again, in designing a set of gears the contact is likely to be 
confined to one pair of teeth in the smaller pinions. 

Each tooth should therefore be made strong enough to sustain 
the whole pressure. 

Rough Teeth. — The teeth of pattern molded gears are apt 
to be more or less irregular in shape, and are especially liable 
to be thicker at one end on account of the draft of the 
pattern. 

In this case the entire pressure may come on the outer corner 
of a tooth and tend to cause a diagonal fracture. 

Let C in Fig. 87 be the point of application of the pressure P, 
and AB the line of probable fracture. 

Drop the 

A_CD on 
AB 

Let AB = x 

and 

CD = y 

angle 

CAD = a 

Fig. 87. 




The bending moment at section ABh, M = Py, and the moment 

of resistance is M' = ^Sxw^ 

5 

where *S = safe transverse strength -y 
of material. 



Py = ^ Sxw^ 




and 



S== 



QPy 



(a) 



If P and w are constant, then iS is a maximum when - is a 

' X 

maximum. 



But y = h sina Siiid X = 



GEAR TEETH 189 

h 



cos a 



- =sina cosa which is a maximum 

X 



y 



when a = 45° and - = ^ 

X ^ 

3P 

Substituting this value in (a) we have S = — ; 

3 P 

But in this case w = A7p and therefore S= ^^^ o 

and p = 3.684 \^ (101) 

diametral pitch, - = .853 V^- (102) 

Unless machine molded teeth are very carefully made, it may 
be necessary to apply this rule to them as well. 

Cut Gears. — With careful workmanship machine molded and 
machine cut teeth should touch along the whole breadth. In 
such cases we may assume a line of contact at crest of tooth and a 
maximum bending moment. 

M = Ph. 

The moment of resistance at base of tooth is 

when h is the breadth of tooth. 

In most teeth the thickness at base is greater than w, but in 
radial teeth it is less. Assuming standard proportions for cut 
gears : 

h = 2lm = .Q7Q5p 
w = .bp 

and substituting above: 

.6765 Pp=^ 

P = MlQbSp. (103) 

For short teeth having h = .bbp formula (103) reduces to: 

P = mbShSp. (104) 

The above formulas are general whatever the ratio of breadth 



190 MACHINE DESIGN 

to pitch. The general practice in this country is to make 

Substituting this value of b in (103) and (104) and reducing: 
Long teeth: p=2.326\/o (105) 

Short teeth: p= 2.098 Vf- (106) 

The corresponding formulas for the diametral pitch are: 

Long teeth: - = 1.35\t (107) 

Short teeth: - = 1.49 \^ (108) 

93. Lewis' Formulas. — The foregoing formulas can only be 
regarded as approximate, since the strength of gear teeth depends 
upon the number of teeth in the wheel; the teeth of a rack are 
broader at the base and consequently stronger than those of a 
pinion. This is more particularly true of epicycloidal teeth. 
Mr. Wilfred Lewis has deduced formulas which take into account 
this variation. For cut spur gears of standard dimensions the 
Lewis formula is as follows: 

P=bSpi.l24:-'-^^) (109) 

where n = number of teeth. 

This formula reduces to the same as (103), for n = 14 nearly. 

Formula (103) would then properly apply only to small pin- 
ions, but as it would err on the safe side for larger wheels, it can 
be used where great accuracy is not needed. The same criticism 
applies to the other formulas in Art. 92. 

The value of S used should depend on the material and on the 
speed. 

The following safe values are recommended for cast iron and 
cast steel. 



GEAR TEETH 



191 



Linear velocity 
ft. per min. 


100 


200 


300 


600 


900 


1200 


1800 


2400 


Cast iron 

Cast steel 


8,000 
24,000 


6,000 
15,000 


4,800 
12,000 


4,000 
10,000 


3,000 
7,500 


2,400 
6,000 


2,000 
5,000 


1,700 
4,250 





For gears used in hoisting machinery where there is slow speed 
and liability of shocks a writer in the Am. Mach. recommends 
smaller values of S than those given above^ and proposes the 
following for four different metals: 



Linear velocity 
ft. per min. 


100 


200 


300 


600 


900 


1200 


1800 


2400 


Gray iron 

Gun metal 

Cast steel 


4,800 

7,200 

9,600 

12,000 


4,200 

6,300 

8,400 

10,500 


3,840 
5,760 
7,680 
9,600 


3,200 
4,800 
6,400 
8,000 


2,400 
3,600 
4,800 
6,000 


1,920 
2,880 
3,840 
4,800 


1,600 
2,400 
3,200 
4,000 


1,360 
2,040 
2,720 
3,400 


Mild steel 





The experiments described in the next article show that the 
ultimate values of S are much less than the transverse strength 
of the material and point to the need of large factors of safety. 

94. Experimental Data. — In the Am. Mach. for Jan. 14, 1897, 
are given the actual breaking loads of gear teeth which failed in 
service. The teeth had an average pitch of about 5 in., a breadth 
of about 18 in. and the rather unusual velocity of over 2000 ft. 
per minute. The average breaking load was aboi^t 15,000 lb. 
there being an average of about 50 teeth on the pinions. Substi- 
tuting these values in (109) and solving we get 

iS = 1575lb. 

This very low value is to be attributed to the condition of 
pressure on one corner noted in Art. 92. Substituting in formula 
for such a case. 



S 



SP 



8150 



.221^2 

This all goes to show that it is well to allow large factors of 
safety for rough gears, especially when the speed is high. 
1 Am. Mach., Feb. 16, 1905. 



192 MACHINE DESIGN 

Experiments have been made by the author on the static 
strength of rough cast-iron gear teeth by breaking them in a test- 
ing machine. The teeth were cast singly from patterns, were 
two pitch and about 6 in. broad. The patterns were constructed 
accurately from templates representing 15 degrees involute 
teeth and cycloidal teeth drawn with a describing circle one-half 
the pitch circle of 15 teeth; the proportions used were those given 
for standard cut gears. 

There were in all 41 cycloidal teeth of shapes corresponding to 
wheels of 15-24-3 6-48-72-1 2Q teeth and a rack. There were 
28 involute teeth corresponding to numbers above given omitting 
the pinion of 15 teeth. 

The pressure was applied by a steel plunger tangent to the 
surface of tooth and so pivoted as to bear evenly across the whole 
breadth. The teeth were inclined at various angles so as to 
vary the obliquity from to 25 degrees for the cycloidal and from 
15 degrees to 25 degrees for the involute. The point of applica- 
tion changed accordingly from the pitch line to the crest of the 
tooth. From these experiments the following conclusions are 
drawn: 

1. The plane of fracture is approximately parallel to line of 
pressure and not necessarily at right angles to radial line through 
center of tooth. 

2. Corner breaks are likely to occur even when the pressure is 
apparently uniform along the tooth. There were fourteen such 
breaks in all. 

3. With teeth of dimensions given, the breaking pressure per 
tooth varies from 25,000 lb. to 50,000 lb. for cycloids as the num- 
ber of teeth increases from 15 to infinity; the breaking pressure 
for involutes of the same pitch varies from 34,000 lb. to 80,000 lb. 
as the number increases from 24 to infinity. 

4. With teeth as above the average breaking pressure varies 
from 50,000 lb. to 26,000 lb. in the cycloids as the angle changes 
from degrees to 25 degrees and the tangent point moves from 
pitch line to crest; with involute teeth the range is between 64,000 
and 39,000 lb. 

5. Reasoning from the figures just given, rack teeth are about 
twice as strong as pinion teeth and involute teeth have an advan- 
tage in strength over cycloidal of from 40 to 50 per cent. The 



GEAR TEETH 193 

advantage of short teeth in point of strength can also be seen. 
The modulus of rupture of the material used was about 36,000 lb. 
Values of >S calculated from Lewis' formula for the various tooth 
numbers are quite uniform and average about 40,000 lb. for 
cycloidal teeth. Involute teeth are to-day generally preferred 
by manufacturers. 

95. Modern Practice. — Two tendencies are quite noticeable 
to-day in the practice of American manufacturers, one toward the 
use of shorter teeth and the other toward a larger angle of 
obliquity. 

The effect of these two changes upon the action of gear teeth 
is the subject of a comprehensive paper read by Mr. R. E. 
Flanders in 1908.^ 

Those readers who desire a detailed mathematical discussion 
of these points are referred to Mr. Flander's paper. 

In brief, the effects of shorter teeth are: (a) To reduce the 
evils of interference with involute teeth; (6) to diminish the arc 
of action; (c) to increase the strength; (d) to increase the durabil- 
ity; {e) to reduce the price of the gear. 

The effects of an increase in the angle of obliquity are (/) 
to diminish interference; (g) to diminish the arc of action; (h) 
to strengthen the teeth; (/) to increase side pressure on bearings; 
(k) to increase the lost work; (l) to distribute the wear more 
evenly. 

It will be noticed from the above that the effects of the two 
changes are mainly the same. To reduce interference, to 
strengthen the teeth, and to secure durability and uniform wear 
are all desirable and important. 

The question of side pressure is not important nor that of lost 
work. The efficiency of accurately cut spur gearing, according 
to experiments by Lewis and others, is between 95 and 98 per 
cent. 

Some examples of modern practice will show the present 
tendencies. (See next page.) 

Mr. Flanders states that seven out of eleven automobile 
manufacturers questioned are using the stub form of tooth and 
like it. 

' Trans. A. S. M. E., 1908. 



194 



MACHINE DESIGN 



• Name of firm 


Involute teeth 


Remarks 


Addendum 


Pressure 
angle 


Wm. Sellers & Co 


0.942 m 
0.785 m 
. 785 m 
/0.70 m 
\ 0.80 m 


20 degrees 
14^ degrees 
20 degrees 

20 degrees 


Steel mill. 


C. W. Hunt Co 


Wellman-Seaver-Morgan Co. 
Fellows Gear Shaper Co ... . 



Note. — m = module = -• 

A committee has recently been appointed by the American 
Society of Mechanical Engineers to investigate the subject of 
interchangeable involute gearing and if practicable to recommend 
a standard form. Mr. Wilfred Lewis, the chairman of the com- 
mittee, is on record as approving a pressure angle of 22^ degrees 
and an addendum of 0.875 m.^ 

Mr. Fellows, another member of the committee, recommends 
an angle of 20 degrees and a = 0.75 m. 

Either of these plans will do away with interference and allow 
of the use of 12-tooth pinions. 

Mr. Gabriel of the Brown & Sharpe Manufacturing Company is 
in favor of retaining the present angle of 14^ degrees and a = m. 

One objection to the present standard is that it is necessary to 
empirically modify the addendum near the crest of the tooth to 
prevent interference. It is claimed on the other hand that 
this " easing off '^ of the point of the tooth is a help in bringing the 
teeth together without shock. 

Teeth cut with a milling cutter or planed by a form can be 
made of empirical shape without difficulty, but teeth generated 
or "hobbed" must correspond in all ways to some theoretical 
curve which matches the rack used as a basis for the system. 

At the present writing, the problem of choosing an acceptable 
standard for involute gears seems far from solution. 

^ Trans. A. S. M. E., 1910. 



« 



GEAR TEETH 195 

96. Teeth of Bevel Gears. — There have been many formulas 
and diagrams proposed for determining the strength of bevel 
gear teeth, some of them being very complicated and incon- 
venient. It will usually answer every purpose from a practical 
standpoint, if we treat the section at the middle of the breadth 
of such a tooth as a spur wheel tooth and design it by the foregoing 
formulas. The breadth of the teeth of a bevel gear should be 
about one-third of the distance from the base of the cone to the 
apex. 

One point needs to be noted; the teeth of bevel gears are 
stronger than those of spur gears of the same pitch and number of 
teeth since they are developed from a pitch circle having an ele- 
ment of the normal cone as a radius. To illustrate, we will sup- 
pose that we are designing the teeth of a miter gear and that the 
number of teeth is 32. In such a gear the element of normal cone 

is V 2 times the radius. The actual shape of the teeth will then 

correspond to those of a spur gear having 32\/2 = 45 teeth nearly. 

Note. — In designing the teeth of gears where the number is unknown, the 
approximate dimensions may first be obtained by formula (105) or (106) and 
then these values corrected by using Lewis' formula. 

PROBLEMS 

1. The drum of a hoist is 8 in. in diameter and makes 5 revolutions per 
minute. The diameter of gear on the drum is 36 in. and of its pinion 6 in. 
The gear on the countershaft is 24 in. in diameter and its pinion is 6 in. in 
diameter. The gears are all cut. 

Calculate the pitch and number of teeth of each gear, assuming a load of 
two tons on drum chain and 5 ==6000. Also determine the horse-power of 
the machine. 

2. Calculate the pitch and number of teeth of a cut cast-steel gear 10 in. 
in diameter, running at 350 revolutions per minute and transmitting 20 
horse-power. 

3. A cast-iron gear wheel is 30 ft. 6f in. in pitch diameter and has 192 
teeth, which are machine-cut and 30 in. broad. 

Determine the circular and diameter pitches of the teeth and the horse- 
power which the gear will transmit safely when making 12 revolutions per 
minute. 

4. A two-pitch cycloidal tooth, 6 in. broad, 72 teeth to the wheel, failed 
under a load of 38,000 lb. Find value of S by Lewis' formula. 

5. A vertical water-wheel shaft is connected to horizontal head shaft by 
cast-iron gears and transmits 150 horse-power. The water-wheel makes 
200 revolutions per minute and the head shaft 100. 



196 MACHINE DESIGN 

Determine the dimensions of the gears and teeth if the latter are approxi- 
mately two pitch. 

6. Work Problem 1, uisng short teeth instead of standard. 

97. Rim and Arms. — The rim of a gear, especially if the teeth 
are cast, should have nearly the same thickness as the base of 
tooth, to avoid cooling strains. 

It is difficult to calculate exactly the stresses on the arms of 
the gear, since we know so little of the initial stress present, due 
to cooling and contraction. A hub of unusual weight is liable 
to contract in cooling after the arms have become rigid and cause 
severe tension or even fracture at the junction of arm and hub. 

A heavy rim on the contrary may compress the arms so as 
actually to spring them out of shape. Of course both of thes3 
errors should be avoided, and the pattern be so designed that 
cooling shall be simultaneous in all parts of the casting. 

The arms of spur gears are usually made straight without 
curves or taper, and of a flat, elliptical cross-section, which offers 
little resistance to the air. To support the wide rims of bevel 
gears and to facilitate drawing the pattern from the sand, the arms 
are sometimes of a rectangular or T section, having the greatest 
depth in the direction of the axis of the gear. For pulleys which 
are to run at a high speed it is important that there should be no 
ribs or projections on arms or rim which will offer resistance to the 
air. Experiments by the writer have shown this resistance to be 
serious at speeds frequently used in practice. 

A series of experiments conducted by the author are reported 
in the Am. Mach. for Sept. 22, 1898, to which paper reference 
is here made. 

Twenty-four pulleys having 3^ in. face and diameters of 16, 
20 and 24 in. were broken in a testing machine by the pull of a 
steel belt, the ratio of the belt tensions being adjusted by levers 
so as to be two to one. Twelve of the pulleys were of the ordinary 
cast-iron type having each six arms tapering and of an elliptic 
section. The other twelve were Medart pulleys with steel rims 
riveted to arms and having some six and some eight arms. Test 
pieces cast from the same iron as the pulleys showed an average 
modulus of rupture of 35,800 for the cast iron and 50,800 for the 
Medart. 

In every case the arm or the two arms nearest the side of the belt 



PULLEY ARMS 197 

having the greatest tension, broke first, showing that the torque 
was not evenly distributed by the rim. Measurements of the 
deflection of the arms showed it to be from two to six times as 
great on this side as on the other. The buckling and springing of 
the rim was very noticeable especially in the Medart pulleys. 

The arms of all the pulleys broke at the hub showing the 
greatest bending moment there, as the strength of the arms at the 
hub was about double that at the rim. On the other hand, some 
of the cast-iron arms broke simultaneously at hub and rim, 
showing a negative bending moment at the rim about one-half 
that at the hub. 

The following general conclusions are justified by these 
experiments: 

(a) The bending moments on pulley arms are not evenly 
distributed by the rim, but are greatest next the tight side of 
belt. 

(b) There are bending moments at both ends of arm, that at 
the hub being much the greater, the ratio depending on the 
relative stiffness of rim and arms. 

The following rules may be adopted for designing the arms of 
cast-iron pulleys and gears: 

1. Multiply the net turning pressure, whether caused by belt 
or tooth, by a suitable factor of safety and by the length of the 
arm in inches. Divide this product by one-half the number of 
arms and use the quotient for a bending moment. Design the 
hub end of arm to resist this moment. 

2. Make section modulus at the rim ends of arms one-half as 
strong as at the hub ends. 

98. Sprocket Wheels and Chains. — Steel chains connecting 
toothed wheels afford a convenient means of getting a positive 
speed ratio when the axes are some distance apart. There are 
three classes in common use, the block chain, the roller chain 
and the so-called '^silent" chain. 

Mr. A. Eugene Michel publishes quite a complete discussion 
of the design of the first two classes in Mchy., for February, 
1905, and reference is here made to that journal. 

Block chain is that commonly used on bicycles and small 
motor cars, so named from the blocks with round ends which are 



198 



MACHINE DESIGN 



used to fill in between the links. The sprocket teeth are spaced 
to a pitch greater than that of the chain links and the blocks 
rest on flat beds between the teeth, Fig. 89. 

Roller chains have rollers on every pin and have inside and 
outside links. The sprocket teeth have the same pitch as the 
chain links, the rollers fitting circular recesses between the 
sprockets, Fig. 90. 

The most serious failing of the chain is its tendency to stretch 
with use so that the pitch becomes greater than that of the 
sprocket teeth. 

To obviate this difficulty in a measure considerable clearance 
should be given to the sprocket teeth as indicated in Fig. 90 
As the pitch of the chain increases it will then ride higher upon 




Fig. 89. 



Fig. 90. 



the sprockets until the end of the tooth is reached. The teeth 
are rounded on their side faces, that they may easily enter the 
gaps in the chain and have side clearance. 

Mr. Michel gives the following values for the tensile strength 
of chains as determined by actual tests. 

Roller Chain 



Pitch inches. 
Tensile 
strength lb . 


1 
1,200 


1,200 


3 

4 

4,000 


1 
6,000 


9,000 


12,000 


J^4 

19,000 


2 
25,000 



Block Chain 

1 inch pitch 1200 to 2500 lb. 
IJ inch pitch 5000 lb. 



CHAIN DRIVES 199 

Mr. Michel further recommends a factor of safety of from 
5 to 40 according to the severity of the conditions as to speed and 
shocks. 

The tendency is to use short links and double or triple width 
chains to increase the rivet bearing surface, as it is this latter 
factor which really determines the life of a chain. 

Roller chains may be used up to speeds of 1000 to 1200 ft. 
per minute. 

The sprocket should be so designed that one tooth will carry 
the load safely with the pressure near the crest since these con- 
ditions obtain as the chain stretches. Use values of *S as in 
Art. 93. 

99. Silent Chains. — The weak points in the ordinary chain, 
whether it be made with blocks or rollers, are the rivet bearings. 
It is the continual wear of these, due to insufficient area and 
lack of proper lubrication, that shortens the life of a chain. 

The so-called '^ silent chain" 
with rocker bearings, is com- 
paratively free from this defect. 
Fig. 91 illustrates the shapes 
of links, rivets and sprockets for 
this kind of chain as manufac- 
tured by the Morse Chain Com- 
pany. 

The chain proper is entirely 
outside of the sprocket teeth so that the latter may be contin- 
uous across the face of the wheel, save for a single guiding 
groove in the center. 

Projections on the under side of the links engage with the 
teeth of the sprocket, E being the point of contact for the driver 
and / a similar point for the follower when the rotation is as 
indicated. 

Each rivet consists practically of two pins called by the makers 
the rocker pin and the seat pin. Each pin is fastened in its 
particular gang of links and the relative motion is merely a 
rocking of one pin on the other without appreciable friction. 

The pins are of hardened tool steel with softened ends. The 
combination of this freedom from rubbing contact with the adap- 




200 



MACHINE DESIGN 



tation of the engaging tooth profiles, gives a chain which can be 
safely run at high speeds without objectionable vibration or 
appreciable wear. 

The chains can be made of almost any width from J in. up to 
18 in., the width depending upon the pitch of the chain and the 
power to be transmitted. 

The following are the working loads (and limiting speeds) of 
chains 2 in. in width and of different pitches, taken from a table 
published by the makers: 



Pitch in inches 


i 


f 


3 


.9 


1.2 


1.5 


Working load in pounds ..... 


130 


190 


236 


380 


520 


760 


Limiting speed revolutions 
per minute. 


2,000 


1,600 


1,200 


1,100 


800 


600 



The number of teeth in the small sprocket may vary frorn 15 
to 30 according to the conditions. 

Assuming 17 teeth and the number of revolutions given in the 
above table the speed of chain would be 1420 ft. per minute for 
the i-in. pitch and 1275 ft. per minute for the 1.5 in. 

Chains of this character have been run successfully at 2000 ft. 
per minute. 



PROBLEMS 

1. Design eight arms of elliptic section for a gear 54. in. pitch diameter, 
to transmit a pressure on tooth of 800 lb. Material, cast iron having a work- 
ing transverse strength of 6000 lb. per square inch. 

2. Two sprocket wheels of 75 and 17 teeth respectively are to transmit 
25 horse-power at a chain speed of about 800 ft. per minute, with a factor of 
safety of 12 — 

Determine the proper pitch of roller chain, the pitch diameters of the 
sprockets, and the numbers of revolutions. 

3. Suppose that in Problem 2, a " silent " chain is to be used and the chain 
speed increased to 1200 ft. per minute. Determine the proper pitch of chain 
to be used if the width of chain is 3 in. Determine diameters and revolutions 
of sprockets as before. 



100. Cranks and Levers. — A crank or rocker arm which is 
used to transmit a continuous or reciprocating rotary motion is in 



CRANKS AND LEVERS 



201 



the condition of a cantilever or bracket with a load at the outer 
end. 

If the web of the crank is of uniform thickness theory requires 
that its profile should be parabolic for uniform strength, the 
vertex of the parabola being at the load point. 

A convenient approximation to this shape can be attained by 
using the tangents to the parabola at points midway between the 




Fig. 92. 



hub and the load point. See Fig. 92. The crank web is designed 
of the right thickness and breadth to resist the moment at AB, 
and the center line is produced to Q, making PQ=hPO. 

Straight lines drawn from Q to A and B will be tangent to the 
parabola at the latter points and will serve as contour lines for 
the web. 

Assume the following dimensions in inches: 

Z = length of crank = OP 

i = thickness of web 

/i = breadth of weh = AB 

d = diameter of eye = cd 
d^ = diameter of pin 

b = breadth of eye 
D = diameter of hub = CD 
2)j= diameter of shaft 
B = breadth of hub. 

If the pressure on the crank pin is denoted by P then will the 

PI 
moment at A 5 be ^ and the equations of moments for the cross- 



section will be: 



Pl^Sth' 
2 ~ 



6 [See Formula (3)] 

and from this the dimensions at A 5 may be calculated. 



202 MACHINE DESIGN 

The moment at the hub will be PI and will tend to break the 
iron on the dotted lines CD. The equation of moments for the 
hub is therefore: 

Pl==^(D'-D,') 

From this equation the dimensions of the hub may be calcu- 
lated when D^ is known. The eye of a crank is most likely to 
break when the pressure on the pin is along the line OP, and the 
fracture will be along the dotted lines cd. The bending moment 
will be P multiplied by the distance from center of pin to center 
of eye measured along axis of pin. If we call this distance x, 
then will the equation of moments be: 

Px = -w-{d — di) 

It is considered good practice among engine builders to make 
the values of x, b and B as small as practicable, in order to reduce 
the twisting moment on the web of the crank and the bending 
moment on the shaft. In designing the hub, allowance must be 
made for the metal removed at the key-way. 

PROBLEM 

Design a cast-steel crank for a steam engine having a cylinder 12 by 30 
in. and an initial steam pressure of 120 lb. per square inch of piston. The 
shaft is 6 in. and the crank pin 3 in. in diameter. The distance x may be 
assumed as 4 in. Calculate, 

1. Dimensions of web at AB. 

2. Dimensions of hub allowing for a key 1 X | in. 

3. Dimensions of eye for pin and make a scale drawing in ink showing 
profile of crank complete. S may be assumed as 6000 lb. per square inch. 

REFERENCES 

Modern American Machine Tools, Benjamin. Chapter VIII. 
Proportions of Arms and Rims. Am. Mach., Sept. 30, 1909. 
Efficiency of Gears. Am. Mach., Jan. 12, 1905; Aug. 19, 1909. 
Strength of Gear Teeth. Mchy., Jan., 1908; Am. Mach., Feb. 16, 1905; 

May 9, 1907; Jan. 16, 1908. 
Proposed Standard Systems of Gear Teeth. Am. Mach., Feb. 25, 1909; 

July 1, 1909. 
Roller Chains. Mchy., Feb., 1905. 
Tests of Short Bearings in Chains. Am. Mach., Dec. 28, 1905. 



CHAIN DRIVES 203 

Progress in Chain Transmission. Am. Mach., Nov. 7, 1907. 
English Chain Drives. Cass., May, 1908. 

Lewis' Experiments on Gears. Tr. A. S. M. E., Vol. VII, p. 273. 
Strength of Gear Teeth. Tr. A. S. M. E., Vol. XVIII, p. 766. 
Chain Gearing. Tr. A. S. M. E., Vol. XXIII, p. 373. 
Symposium on Gearing. Tr. A. S. M. E., Vol. XXXII, p. 807. 



CHAPTER XI 




FLY-WHEELS 

101. In GeneraL — The hub and arms of a fly-wheel are designed 
in much the same way as those of pulleys and gears, the straight 
arm with elliptic section being the favorite. The rims of such 
wheels are of two classes, the wide, thin rim used for belt trans- 
mission and the narrow solid rim of the generator or blowing 

engine wheel. Fly-wheels up to 
8 or 10 ft. in diameter are usually 
cast in one piece; those from 10 
to 16 ft. in diameter may be cast 
in halves, while wheels larger 
than the last mentioned should 
be cast in sections, one arm to 
Fjq 93 each section. 

This is a matter, not of use, 
but of convenience in casting and in transportation. 

The joints between hub and arms and between arms and rim 
need not be specially considered here, since wheels rarely fail 
at these points. 

The rim and the joints in the rim cannot be too carefully 
designed. The smaller wheel 
cast in one piece is more or 
less subject to stresses caused 
by shrinkage. The sectional 
wheel is generally free from 
such stresses but is weakened 
by the numerous joints. 

Rim j oints are of two gen- 
eral classes according as bolts 
or links are used for fastenings. 

Wide, thin rims are usually fastened together by internal 
flanges and bolts as shown in Fig. 93, while the stocky rims of 
the fly-wheels proper are joined directly by links or T-head 
"prisoners" as in Fig. 94. 

204 




Fig. 94. 



FLY WHEELS 205 

As will be shown later, the former is a weak and unreliable 
joint, especially when located midway between the arms. 

The principal stresses in fly-wheel rims are caused by centrifu- 
gal force. 

102. Safe Speed for Wheels. — The centrifugal force developed 
in a rapidly revolving pulley or gear produces a certain tension 
on the rim, and also a bending of the rim between the arms. 
We will first investigate the case of a pulley having a rim of uni- 
form cross-section. 

It is safe to assume that the rim should be capable of bearing 
its own centrifugal tension without assistance from the arms. 

Let D = mean diameter of pulley rim 
i = thickness of rim 
h = breadth of rim 

1^ = weight of material per cubic inch 
= .26 lb. for cast iron 
= . 28 lb. for wrought iron or steel 
n = number of arms 
A^ = number revolutions per minute 
?; = velocity of rim in feet per second. 

First let us consider the centrifugal tension alone. The cen- 
trifugal pressure per square inch of concave surface is 

P= (a) 

gr 

where W is the weight of rim per square inch of concave surface 

= .e,and. = racliusinfeet = #. 
' 24 

The centrifugal tension produced in the rim by this force is 

by formula (15) 

Substituting the values of p, W and r and reducing : 

^ = 1^^ (110) 

and v = ^l^- (111) 



206 MACHINE DESIGN 

For an average value of w = .27, {S9) reduces to 

S = -ynT nearly 

a convenient form to remember. 

The corresponding values of *S for dry wood and for leather 
would be nearly: 

Wood S 



100 

Leather '^ = orS' 

o\) 

If we assume S as the ultimate tensile strength, 16,500 lb. 
for cast iron in large castings and 60,000 lb. for soft steel, then 
the bursting speed of rim is: 

for a cast-iron wheel ?; = 406 ft. per second (112) 

and for steel rim v = nb ft. per second (113) 

and these values may be used in roughly calculating the safe 
speed of pulleys. 

It has been shown by Mr. James B. Stanwood, in a paper read 
before the American Society of Mechanical Engineers,^ that each 
section of the rim between the arms is moreover in the condition 
of a beam fixed at the ends and uniformly loaded. 

This condition will produce an additional tension on the outside 
of rim. The formula for such a beam when of rectangular cross- 
section is 

Wl Sb(P 

W in this case is the centrifugal force of the fraction of rim 
included between two arms. 

The weight of this fraction is and its centrifu2;al force 

TzDUw 24^2 ^^^ 2^nhtwv^ 

W= X—F^ or W= 

n gD gn 

Also I = — and d = t 

n 

» See Trans. A. S, M. E., Vol. XIV. 



FLY WHEELS 207 



Substituting these values in (b) and solving for aS : 

tn 
If w is given an average value of .27 then 



>S = 3.678^:;:^ (c) 



S = —^ nearly (d) 

and the total value of the tensile stress on outer surface of rim is 

S'=5^VfQ nearly. (114) 

Solving for v : 

v = 



VI 



P_ 1' (115) 

^^2 + 10 

In a pulley with a thin rim and small number of arms, the 
stress due to this bending is seen to be considerable. 

It must, however, be remembered that the stretching of the 
arms due to their own centrifugal force and that of the rim will 
diminish this bending. Mr. Stanwood recommends a deduction 
of one-half from the value of S in (d) on this account. 

Prof. Gaetano Lanza has published quite an elaborate mathe- 
matical discussion of this subject. (See Vol. XVI, Trans. 
A. S. M. E.) He shows that in ordinary cases the stretch of the 
arms will relieve more than one-half of the stress due to bending, 
perhaps three-quarters. 

103. Experiments on Fly-wheels. — In order to determine 
experimentally the centrifugal tension and bending in rapidly 
revolving rims, a large number of small fiy-wheels have been 
tested to destruction at the Case School laboratories. In all 
ten wheels, 15 in. in diameter and twenty-three wheels 2 ft. in 
diameter have been so tested. An account of some of these 
experiments may be found in Trans. A. S. M. E., Vol. XX. 
The wheels were all of cast iron and modeled after actual fly- 
wheels. Some had solid rims, some jointed rims and some steel 
spokes. 

To give to the wheels the speed necessary for destruction, 
use was made of a Dow steam turbine capable of being run at any 
speed up to 10,000 revolutions per minute. The turbine shaft 



208 



MACHINE DESIGN 



was connected to the shaft carrying the fly-wheels by a brass 
sleeve coupling loosely pinned to the shafts at each end in such a 
way as to form a universal joint, and so proportioned as to break 
or slip without injuring the turbine in case of sudden stoppage 
of the fly-wheel shaft. 

One experiment with a shield made of 2-in. plank proved that 
safety did not lie in that direction, and in succeeding experiments 
with the 15-in. wheels a bomb-proof constructed of 6Xl2-in. 
white oak was used. The first experiment with a 24-in. wheel 
showed even this to be a flimsy contrivance. In subsequent 
experiments a shield made of 12 X 12-in. oak was used. This 
shield was split repeatedly and had to be re-enforced by bolts. 

A cast-steel ring about 4 in. thick, lined with wooden blocks 
and covered with 3-in. oak planking, was finally adopted. 

The wheels were usually demolished by the explosion. No 
crashing or rending noise was heard, only one quick, sharp report, 
like a musket shot. 

The following tables give a summary of a number of the 
experiments. 

TABLE L 

Fifteen-inch Wheels 



No. 


Bursting speed 


Centrifugal 
tension 

"10 


Remarks 


Rev. 
per minute 


Feet per 
second =v 


1 

2 
3 
4 
5 
6 
7 
8 
9 
10 


6,525 

6,525 

6,035 

5,872 

2,925 

5,600 ' 

6,198 

5,709 

5,709 

5,709 


430 
430 
395 
380 
192 
368 
406 
368 
365 
361 


18,500 
18,500 
15,600 
14,400 
3,700 
13,600 
16,500 
13,600 
13,300 
13,000 


Six arms. 
Six arms. 
Thin rim. 
Thin rim. 
Joint in rim. 
Three arms. 
Three arms. 
Three arms. 
Thin rim. 
Thin rim. 



Doubtful. 



FLY WHEELS 



209 



TABLE LI 

Twenty-four-inch Wheels 



No. 


Shape and size of rim 


Weight 

of 
wheel, 
pounds 


Diam- 
eter, 
inches 


Breadth 
inches 


Depth, 
inches 


Area, 
square 
inches 


Style of joint 


11 
12 
13 
14 
15 
16 
17 


24 
24 
24 
24 
24 
24 
24 


2| 

4iV 

4 

4 

4i\ 

1.2 

1.2 


1.5 
.75 
.75 

.75 
.75 

2.1 

2.1 


3.18 
3.85 
3.85 
3.85 
3.85 
2.45 
2.45 


Solid rim 


75.25 

93. 

91.75 

95. 

94.75 

65.1 

65. 


Internal flanges, bolted 

Internal flanges, bolted 

Internal flanges, bolted 

Internal flanges, bolted 

Three lugs and links 


Two lugs and links 



TABLE LII 
Flanges and Bolts 



No. 


Flanges 


Bolts 


Thickness, 
inches 


Effective 

breadth, 

inches 


Effective 
area, 
inches 


No. to each 
joint 


Diameter, 
inches 


Total tensile 

strength, 

pounds 


12 
13 
14 
15 


n 

IS 

IS 
in 
To 


2.8 
2.75 
2.75 
2.5 


1.92 
1.89 

2.58 
2.34 


4 
4 
4 
4 


Iff 
t 


16,000 
16,000 
16,000 
20,000 



BY TESTING MACHINE 



Tensile strength of cast iron =19,600 lb. per square inch. 
Transverse strength of cast iron =46,600 lb. per square inch. 
Tensile strength of y5_ bolts = 4,000 lb. 
Tensile strength of | bolts = 5,000 lb. 



210 



MACHINE DESIGN 



TABLE LIII 
Failure of Flanged Joints 



No. 


Area of 

rim, 
square 
inches 


Effect 
area 
flanges, 
square 
inches 


Total 
strength 

bolts, 
pounds 


Bursting 
speed 


Cent, tension 


Remarks 


Rev. 
per 
min. 


Ft. per 
sec. 

= v 


Per 

sq. in. 

10 


Total 
lb. 


11 
12 
13 
14 
15 


3,18 
3.85 
3.85 
3.85 
3,85 






3,672 


385 


14,800 


47,000 


Solid rim. 
Flange broke. 
Flange broke. 
Bolts broke. 
Flange broke. 


1.92 
1.89 
2.58 
2.34 


16,000 
16,000 
16,000 
20,000 


1,760 
1,875 
1,810 


184 
196 
190 


3,400 
3,850 
3,610 


13,100 
14,800 
13,900 



TABLE LIV 
Linked Joints 



No. 


Lugs 


Links 


Rim 


Breadth 
inches 


Length 
inches 


Area, 
sq. in. 


Number 
used 


Effect 
breadth, 
inches 


Thick- 
ness, 
inches 


Effective 
area, 
sq. in. 


Max. 
area, 
sq. in. 


Net 
area, 
sq. in. 


16 
17 


.45 
.44 


1.0 

.98 


.45 
.43 


3 

2 


,57 
,54 


,327 
.380 


,186 
,205 


2.45 

2.45 


1.98 
1.98 



BY TESTING MACHINE 

Tensile strength of cast iron = 19,600. 
Transverse strength of cast iron = 40,400. 
Av. tensile strength of each link = 10, 180. 

TABLE LV 
Failure of Linked Joints 



No. 


Strength 
of links, 
pounds 


Strength 
of rim, 
pounds 


Bursting speed 


Cent, tension 


Remarks 


Rev. per 
min. 


Ft. per 

sec. =tJ 


Per sq. in. 
10 


Total 


. 16 
17 


30,540 
20,360 


38,800 
38,800 


3,060 
2,750 


320 
290 


10,240 
8,410 


25,100 
20,600 


Rim broke. 
Lugs and rim 
broke. 



FLY WHEELS 211 

The flanged joints mentioned had the internal flanges and bolts 
common in large belt wheel rims while the linked joints were such 
as are common in fly-wheels not used for belts. 

Subsequent experiments^ have given approximately the same 
results as those ju&t detailed. The highest velocity yet attained 
has been 424 ft. per second; this is in a solid cast-iron rim with 
numerous steel spokes. The average bursting velocity for solid 
cast rims with cast spokes is 400 ft. per second. 

Wheels with jointed rims burst at speeds varying from 190 to 
250 ft. per second, according to the style of joint and its location. 
The following general conclusions seem justified by these tests. 

1. Fly-wheels with solid rims, of the proportions usual among 
engine builders and having the usual number of arms, have a 
sufficient factor of safety at a rim speed of 100 ft. per second if 
the iron is of good quality and there are no serious cooling strains. 

In such wheels the bending due to centrifugal force is slight, 
and may safely be disregarded. 

2. Rim joints midway between the arms are a serious defect and 
reduce the factor of safety very materially. Such joints are as 
serious mistakes in design as would be a joint in the middle of a 
girder under a heavy load. 

3. Joints made in the ordinary manner, with internal flanges 
and bolts, are probably the worst that could be devised for this 
purpose. Under the most favorable circumstances they have 
only about one-fourth the strength of the solid rim and are par- 
ticularly weak to resist bending. 

See Fig. 95, which shows the opening of such a joint and the 
bending of the bolts. 

In several joints of this character, on large fly-wheels, calcu- 
lation has shown a strength less than one-fifth that of the rim. 

4. The type of joint known as the link or prisoner joint is 
probably the best that could be devised for narrow rimmed 
wheels not intended to carry belts, and possesses, when properly 
designed, a strength about two-thirds that of the solid rim. 

In 1902-04 experiments on four-foot pulleys were conducted 
by the writer, and the results published.^ 

A cast-iron, whole rim pulley 48 in. in diameter, burst at 1100 

1 Trans. A. S. M. E., Vol. XXIII. 

2 Trans. A. S. M. E., Vol. XXVI. 



212 MACHINE DESIGN 

revolutions per minute or a linear speed of 230 ft. per second, 
the rupture being caused by a balance weight of 3^ lb. which had 
been riveted inside the rim by the makers. The centrifugal 
force of this weight at 1100 revolutions per minute was 2760 lb. 

A cast-iron split pulley of the same dimensions burst at a speed 
of about 600 revolutions per minute, or a linear speed of only 125 
ft. per second. 

The failure was due to the unbalanced weight of the joint 
flanges and bolts which were located midway between the arms. 
Such a pulley is not safe at high belt speeds. 

104. Wooden Pulleys. — Experiments on the bursting strength 
of wooden pulleys were conducted at the Case School laboratories 
in 1902-3 under the writer's direction.^ 

These are of some interest in view of the use of this material 
for fly-wheel rims. As noted in Art. 102, the tensile stress in 
wood due to the centrifugal force is only -^^ that of cast iron under 
similar circumstances. Assuming the tensile strength of the 
wood to be 10,000 lb. per square inch, and substituting this value 

in the equation S = j^ we have the burstmg speed of a wooden 

pulley 15 = 1000 ft. per second nearly. 

This for wood without joints. 

The 24-in. pulleys tested had wood rims glued up in the usual 
manner and jointed at two opposite points. The wheels burst 
at speeds varying from 1700 to 2450 revolutions per minute, or 
linear rim speeds varying from 178 to 257 ft. per second, thus 
comparing favorably with cast-iron split pulleys. The rims 
usually failed at the points where the arms were mortised in, and 
the stiffening braces at these points did more harm than good. 
A wooden pulley with solid rim and web remained intact at 
4450 revolutions per minute, or 467 ft. per second, a higher speed 
than that of any cast-iron pulley tried. 

105. Rims of Cast-iron Gears. — A toothed wheel will burst at 
a less speed than a pulley because the teeth increase the weight 
and therefore the centrifugal force without adding to the strength. 

The centrifugal force and therefore the stresses due to the force 

' Mchy., N. Y., Aug., 1905. 



FLY WHEELS 



213 




S 
cu 

o 

w 

H 
'•I 

l-H 

o 

•-5 



o 

H 

CM 

O 



6 



214 



MACHINE DESIGN 



will be increased nearly in the ratio that the weight of rim and 
teeth is greater than the weight of rim alone. 

This ratio in ordinary gearing varies from 1.5 to 1.7. We will 
assume 1.6 as an average value. Neglecting bending we now 
have from equation (110) 



*S-1.6X 



and 



12wv^ 
9 


I9.2w;?;2 
9 


.= .i 


9S 





(116) 



19.2w 
= 326.2 ft. per second 



Including bending 



S'-l.Qv^ 




(117) 



(118) 



As the transverse strength of cast iron by experiment is about 
double the tensile strength, a larger value of S may be allowed in 
formulas (114) (115) (118). 

In built-up wheels it is better to have the joints come near 
the arms to prevent the tendency of the bending to open the 

joints, and the fastenings 
should .have the same tensile 
strength as the rim of the 
wheel. 

106. Rotating Discs.— The 
formulas derived in Art. 102 
will only apply in the case of 
thin rims and cannot be used 
for discs or for rims having 
any considerable depth. The 
determination of the stresses 
in a rotating disc is a com- 
plicated and difficult problem, if the material is regarded as 
perfectly elastic. 

A rational solution of this problem may be found in Stodola's 
Steam Turbines, pp. 157-69. For the purposes of this treatise 
an approximate solution is preferred, the elasticity of the metal 
being neglected. This method of treatment is much simpler, 




U 



Fig. 96. 



ROTATING DISCS 215 

and as the metals used are imperfectly elastic (especially the 
cast metals) the results obtained will probably be as reliable as 
any — for practical use. 

The following discussion is an abstract of one given by Mr. 
A. M. Levin in the Am. Mach.^ the notation being changed 
somewhat. 

107. Plain Discs. — Let Fig. 96 represent a ring of uniform 
thickness t, having an external diameter D and an internal 
diameter d, all in inches. 

Let V = external velocity in feet per second 

Let a = angular velocity =—^ 

r = radius to center of gravity of half ring in feet 
It; = weight of metal per cubic inch. 
The value of r for a half- ring is easily proved to be: 

2 D^-d^ . . , 

^s~ • 1^ — 7^ m inches 
3k D^ — d^ 

or 1 D^-d^ 



The weight of the half-ring is: 



in feet. 






and its centrifugal force : 



^ Wa'r aHw(D^-d^) ,,,„, 

9 144sr 

Substituting for a its value in terms of v: 

p^4W^. (120) 

Now if we assume the stress on the area at AB due to the 
centrifugal force to be uniformly distributed: (and here lies the 
approximation) then will the tensile stress on the section be 

^ C _4wv\D^ + Dd+d') ,^„^, 

* Am. Mach., Oct. 20, 1904. 



216 



MACHINE DESIGN 



For a solid disc: 



For a thin ring: 



Sd=o 



Sd=T) — 



4:WV' 

9 
9 



(122) 



(123) 



or the same as in equation (110). 

If the metal be perfectly elastic, Stodola's formulas give 

uWV 

S= as the stress near the center when d approaches — 

or more than twice the value given in (122). In view of the 
imperfect elasticity of the metals used the true value will prob- 
ably be between these two. This value should be determined 
by experiment. 

108. Conical Discs. — Let Fig. 97 represent a ring whose thick- 
ness varies uniformly from the inner to the outer circumference 
and whose dimensions are as follows : 




Fig. 97. 

D = outer diameter in inches 

d = inner diameter in inches 

6 = breadth of ring at inner circumference 
m = tangent of angle of slant CAD 



Then m 



D-d 



or h 



D-d 

m 



ROTATING DISCS 217 

By cutting the ring into slices perpendicular to the axis, 
finding the centrifugal force for each slice and then integrating 
between D and d, the centrifugal force of the half-ring is found 
to be: 

The area on the line AJ5 to resist the centrifugal force is: 

{D-dy , ^ 2wv\D' + M^-4.Dd^) ,^^^. 

-^MT "^^ ^ = gD\D-dY -' ^125) 

Whend = 0: 

5 = ?^. (126) 

or a stress one-half that of a plain flat disc. 

109. Discs with Logarithmic Profile. — A form of disc some- 
times used for steam^ turbines consists of a solid of revolution 
generated by a curve of the equation 

revolving around the x— axis. 

Mr. Levin investigates two curves of this character: 

X 

y = \og X and y=2 log ^ 

o 

and finds the stresses to be respectively: 

Whena = 6 ^ = 1.5^^. (127) 

9 



wv 



When a = §6 S = 1.2 — . (128) 

The general equation for S in this case is: 



S = 96^-"4 (129) 

and in deriving the formulas (127) and (128) D is assumed 
as 8a and as 9a respectively. 



218 



MACHINE DESIGN 



110. Bursting Speeds. — It will be seen that all the formulas 
for centrifugal stress may be reduced to the general form: 

,2 



S = k 



9 



(130) 



where A; is a constant depending upon the shape of the rotating 
body. 

_ fgs 



The following table gives the values of v 



kw 



, the burst- 



ing speed of rim in feet per second, for different materials 
and different shapes. 

TABLE LVI 

Bursting Speeds in Feet per Second 



Metal 


Weight 

per 

cubic 

inch 


Tensile 
strength 


Values of v 


Thin 
ring 


Perforated 

disc 
(Stodola) 


Flat 
disc 


Taper 
disc 


Logar- 
ithmic 
disc 




w 


S 


k = 12 


A; = 9 


A; = 4 


k = 2 


A; = 1.5 


Cast iron 

Manganese bronze .... 
Soft steel 


.26 
.315 

.28 


18,000 
60,000 
60,000 


430 
715 
760 


500 

825 
880 


745 
1,240 
1,315 


1,050 
1,750 
1,860 


1,215 
2,050 
2,140 







111. Tests of Discs. — During the years 1906-07, certain tests 
were made on cast-iron discs in the laboratories of the Case 
School of Applied Science by senior students, Messrs. Baxter, 
Brown, Goss and Jeffrey. 

The discs experimented upon were from 16 to 18 in. in diameter 
and from ^ to 1 in. in thickness and were cast from a soft gray 
iron, clean and free from defects. The average tensile strength 
of the iron was 15,750 lb. per square inch and the transverse 
strength 37,800 lb. per square inch. All of the discs were finished 
to insure good balancing. 

They were tested to bursting by centrifugal force with the 
apparatus before described in the article on Fly-wheels, the 
speed being measured by a reducing gear and counter. Each of 
the discs had a 1-in. hole through the center; some had hubs 2 in. 



ROTATING DISCS 



219 



in diameter and some were plain as noted in the following table 
which gives a resume of the results. 

TABLE LVII 

Bursting Speed of Cast-iron Discs 



Diameter, 
inches 


Weight, 
pounds 


Thickness, 
inches 


Length of 

hub, 

inches 


Bursting 
speed 
r.p.m. 


Calculated 


Velocity of 
rim, ft. per sec. 




18 
18 
18 
16 
16 
16 
16 
18 
18 
16 
16 
18 




.418 
.775 
.573 
.562 
.514 
1.086 
.951 
.953 
.715 
.820 
.873 
.961 


2.00 

2.00 

2.00 

2.125 

2.125 

None 

None 

None 

2.00 

1.57 

1.62 

1.85 


7,755 
7,125 
8,700 
9,282 
9,486 
9,690 
8,262 
8,364 
9,180 
8,874 
9,792 
9,792 


610 
560 
683 
650 
660 
676 
577 
656 
720 
620 
685 
770 


5.25 
6.24 
4.19 
4.62 
4.49 
4.28 
5.86 
4.55 
3.76 
5.08 
4.16 
3.29 






28.75 
27.25 
55.50 
48.00 
61.25 
47.75 
42.00 
45.00 
65.50 



Average value of ^ = 4.64. 

The presence or absence of a hub has no apparent effect on the strength. The value of 
A;, as was to have been expected is slightly greater than the 4 given in formula (122). 



PROBLEMS 

1. Determine bursting speed in revolutions per minute, of a gear 42 in. 
in diameter with six arms, if the thickness of rim is . 75 in. 

(1) Considering centrifugal tension alone. 

(2) Including bending of rim due to centrifugal force assuming that 
three-fourths the stress due to bending is reheved by the stretching of the 
arms. 

2. Design a link joint for the rim of a fly- wheel, the rim being 8 in. wide, 
12 in. deep and 18 ft. mean diameter, the links to have a tensile strength of 
65,000 lb. per square inch. Determine the relative strength of joint and the 
probable bursting speed. 

3. Discuss the proportions of one of the following wheels in the laboratory 
and criticise dimensions. 

(a) Fly-wheel, Allis engine. 

(b) Fly-wheel, Fairbanks gas engine. 

(c) Fly-wheel, air compressor. 

(d) Fly-wheel, Buckeye engine. 

(e) Fly-wheel, pumping engine. 

4. Determine the value of C in formula (124) by calculation. 

5. A Delaval turbine disc is made of soft steel in the shape of the logarith- 



220 MACHINE DESIGN 

mic curve without any hole at the center. Determine the probable bursting 
speed if the disc is 12 in. in diameter. 

6. A wheel rim is made of cast iron in the shape of a ring having diameters 
of 4| ft. and 6 ft., inside and outside. Determine probable bursting speed. 

7. Substitute the value for centrifugal force in place of internal pressure 
in Barlow's formula (b) Art. 22, and derive a value for *S in a rotating ring. 

Test this for d = ^ and compare with formulas in preceding article. 



REFERENCES 

Resistance of Air to Fly-wheels. Cass., Feb., 1903; Mar., 1903. 

Shields to Reduce Windage. Power, Dec, 1903. 

Testing Pit at Purdue University. Am. Mach., Aug, 26, 1909. 

High Efficiency Joint for Rim. Am. Mach., Feb. 28, 1907; Apr. 4, 1907; 

Apr. 11, 1907. 
Rolling Mill Fly-wheels, Tr. A, S. M, E,, Vol. XX, p, 944. 



CHAPTER XII 
TRANSMISSION BYi BELTS AND ROPES 

112. Friction of Belting. — The transmitting power of a belt is 
due to its friction on the pulley, and this friction is equal to the 
difference between the tensions of the driving and slack sides for 
the belt. 

Let It; = width of belt 

T^ = tension of driving side 



^2 = tension of slack side 



C< M 



f 



"-C 




Fig. 98. 



jR= friction of belt 

= T —T 
f= coefficient of friction between 

belt and pulley 
(9 = arc of contact in circular 
measure. 

The tension T at any part of the arc of contact is intermediate 
between T^ and T^. 

Let AB Fig. 98 be an indefinitely short element of the arc of 
contact, so that the tensions at A and B differ only by the 
amount dT. 

dT will then equal the friction on AB which we may call dR. 

Draw the intersecting tangents OT and OT' to represent the 

tensions and find their radial resultant OP. Then will OP 

represent the normal pressure on the arc AB which we will call P. 

<OTP=<ACB = dd 



The friction on A 5 is 



or 



and 



.'. P = Tdd. 

fP=fTdd 
dT = dR^fTdd 

MR ^^ 
JdU = -^' 



221 



222 



MACHINE DESIGN 



Integrating for the whole arc d: 

T.dT 



T, 






T = 

■'■ 2 



rp <^ 

^ 2 



-fd 



40 



R=T,-1\=T,{1- e-f^). (131) 

The value of /varies with the nature of the materials used, the 
tension and slip of the belt and the speed of the pulleys. If we 
denote the expression (1 —e—^^) by C, then for different values 
of /and the arc of contact, C has the following values. 

Arc of Contact 



Values of/ 


90 


120 


150 


180 


200 


.20 


.270 


.342 


.408 


.467 


.503 


.25 


.325 


.407 


.480 


.544 


.582 


.30 


.376 


.467 


.544 


.610 


.649 


.35 


.423 


.520 


.600 


.667 


.705 


.40 


.467 


.567 


.649 


.715 


.753 


.45 


.507 


.610 


.692 


.757 


.792 



The friction or force transmitted by a belt per inch of width 
is then 



R = CT, 



(132) 



and T^ must not exceed the safe working tensile strength of the 
material. 

A handy rule for calculating belts assumes C = .5 which means 
that the force which a belt will transmit under ordinary condi- 
tions is one-half its tensile strength. 

The conditions assumed above are only average ones and the 
formulas are only approximate for any particular case. The 
coefficient of friction varies with the materials used for pulleys 
and belts, with the tension, the speed and the amount of slip. 



BELTS 223 

The sum of the tensions is not constant as may be readily proved 
by experiment. Mr. Barth shows from theoretical considerations 
of the elasticity of the belt that approximately:^ 

where To = initial tension (unloaded), or that the sum of the 
square roots of the two tensions is a constant. For instance, if 
To = 100, we have 

\/t\+\/¥,=20 
If we assume values for T^ and solve for T'2, we have: 



T, 


T 

^ 2 


T, + T, 


121 


81 


202 


144 


64 


208 


169 


49 


218 


196 


36 


232 



and the sum of the tensions increases as the load increases. 

113. Slip of Belt. — Mr. Wilfred Lewis in his experiments on 
belts^ found that the coefficient of friction varied with the slip, 
increasing as the slip increased, so that as the load became 
heavier the slipping of the belt increased its driving power and 
prevented further slip. 

A distinction must be made between slip due to the load and 
slip, or " creep " as it is usually called, due to the stretching of the 
belt. 

As has been already explained, the tension of a belt varies in 
passing over the driving pulley from T^ to Tg and in passing over 
the driven pulley from T'2 to T^^. The belt is elastic and stretches 
more or less according to the tension, so that its length is con- 
tinually changing as it passes over either pulley. This produces 
a " creep " or relative motion of the belt on the pulley, positive 
on one pulley and negative on the other; i.e., the belt gains on the 
driven pulley and loses on the driving pulley. 

Experiments by Professor Bird^ show a creep under ordinary 

' Trans. A. S. M. E., 1909. 

2 Trans. A. S. M. E., 1886. 

3 Trans. A. S. M. E., 1905. 



224 MACHINE DESIGN 

conditions of about 1 per cent and a working modulus of elasticity 
for leather belting of from 12,000 to 30,000 with an average of 
20,000. Slip due to increase of load will be added to the creep. 

Tests of belting reported in 1911 by Professor W. M. Sawdon^ 
indicate a marked variation in the slip of belts without any 
apparent change in the conditions. 

With the belt tension, the load and the speed remaining the 
same, the slip would sometimes remain constant at 1 or 2 per 
cent for 30 or 35 minutes and then suddenly rise to 10 or 15 per 
cent. 

In these experiments it was found that the load capacity of a 
leather belt on pulleys of various materials was as follows, cast 
iron being taken as a standard: 

Cast iron 100 

Wood 105 

Paper 137 

The effect of cork inserts was to increase the driving capacity 
of the cast-iron pulleys 10 to 12 per cent. The wood pulleys 
received no benefit from cork inserts while the capacity of the 
paper pulleys was diminished by the cork. 

The wood pulleys showed a small overload capacity, being 
inferior to the cast iron at slips exceeding 3 or 4 per cent. 

114. Coefficient of Friction. — ^Mr. Barth, as a result of the 
experiments of Mr. Lewis and an exhaustive study of the whole 
subject, suggests the following formulas for the coefficient: 

/=0-54-5^ (134) 

where v is the average velocity of sliding of belt on pulley or 
one-half the .total slip in feet per minute and V is the velocity 
of belt in feet per minute. 

1 Proc. Nat. Ass'n of Cotton Manufacturers, Sept., 1911. 



SLIP OF BELTS 



225 



of / from equation 


(133) are: 


/= 




2 0.267 


See Ai 


4 0.350 




6 0.400 




8 0.433 





112. 



Values of / from equation (134) are: 

y= f= 

400 0.384 

800 0.432 

1600 0.473 

3200 0.512 

It will be noted that these values of / are larger than that 
assumed in Art. 112 and furthermore that some definite relation 
is assumed to exist between the slip of the belt and its speed. 

Equating the two values of /in (133) and (134) and neglecting 
the difference in the constant terms, we have approximately: 



v=:3.14 + .014y 



(135) 



or a total slip of about 6 ft. per minute plus about 3 per cent of 
the linear velocity. 



115. Strength of Belting. — The strength of belting varies 
widely and only average values can be given. According to 
experiments made by the author good oak tanned belting has a 
breaking strength per inch of width as follows: 





Single 


Double 


Solid leather 


900 lb. 
600 lb. 


1,400 lb. 
1,200 lb. 


Where riveted 


Where laced 


350 lb. 





Canvas belting has approximately the same strength as 
leather. Tests of rubber coated canvas belts 4-ply, 8 in. wide, 
show a tensile strength of from 840 lb. to 930 lb. per inch of width. 



226 MACHINE DESIGN 

116. Taylor's Experiments. — The experiments of Mr. F. W. 
Taylor, as reported by him in Trans. A. S. M. E., Vol. XV, afford 
the most valuable data now available on the performance of belts 
in actual service. 

These experiments were carried on during a period of nine years 
at the Midvale Steel Works. Some of Mr. Taylor's conclusions 
are as follows: 

1. Narrow double belts are more economical. than single ones 
of a greater width. 

2. All joints should be spliced and cemented. 

3. The most economical belt speed is from 4000 to 4500 ft. 
per minute. 

4. The working tension of a double belt should not exceed 35 
lb. per inch of width, but the belt may be first tightened to about 
double this. 

5. Belts should be cleansed and greased every six months. 

6. The best length is from 20 to 25 ft. between centers. 

117. Rules for Width of Belts.— It will be noticed that Mr. 
Taylor recommends a working tension only -f^ to ^^ the breaking 
strength of the belt. He justifies this by saying that belts so 
designed gave much less trouble from stoppage and repairs and 
were consequently more economical than those designed by the 
ordinary rules. 

It must be remembered that a belt which is strained to an exces- 
sive tension will not retain this tension long, but will stretch 
until the tension becomes such as the belt will carry comfortably. 

If the belt is under size for the required load this will cause 
slipping and necessitate further tightening and so on. There 
will thus be continual loss of time, so that such a belt is uneco- 
nomical although theoretically of ample strength. 

In the following formulas 50 lb. per inch of width is allowed 
for double belts and 30 lb. for single belts. These are suitable 
values for belts which are not running continuously. The 
formulas may be easily changed for other thicknesses and for 
other values of CT^. 

Let i7P = horse-power transmitted 

Z) = diameter of driving pulley in inches 
iV = number revolutions per minute of pulley. 



WIDTH OF BELTS 227 



The moment of force transmitted by belt is 

RD CT^wD 



2 2 



= T 



Substituting the values assumed for CT^ and solving for w: 

HP 

Single belts ^ = 4200 ^^- (137) 

HP 

Double belts ti; = 2500 jy^- (138) 

The most convenient rules for belting are those which give 
the horse-power of a belt in terms of the surface passing a fixed 
point per minute. 

In formula (136) HP = ^|^^ 

we will substitute the following : 

w 
TF= width of belt m feet = y^ 

K = velocity m ft. per mm. = 



or HP=^ 



12 
lUCT.WV 



1260507? 

Substituting values of C and T^ as before and solving for 
Try = square feet per minute we have approximately: 

Single belts WV = 90HP. (139) 

Double belts WV = 55HP. (140) 

118. Speed of Belting. — As in the case of pulley rims, so in that 
of belts a certain amount of tension is caused by the centrifugal 
force of the belt as it passes around the pulley. 

1 2wv^ 
From equation (110) S = 

where 1; = velocity in feet per second 

It; = weight of material per cubic inch 
>S = tensile stress per square inch. 



228 MACHINE DESIGN 

To make this formula more convenient for use we will make 
the following changes in the contsants: 

Let y = velocity of belt in ft. per minute = 602? 
w = weight of ordinary belting 

= .032 lb. per cubic inch 
^1 = tensile stress per inch width, caused by centrifugal 
force 
= about y^g- *S for single belts. 

V 

Then ^ = ;^ 

dO 

o 16^1 

Substituting these values in (110) and solving for S^ 

72 

'^^^ 1610000 ^^^^^ 

The speed usually given as a safe limit for ordinary belts is 
3000 ft. per minute, but belts are sometimes run at a speed 
exceeding 6000 ft. per minute. 

Substituting different values of V in the formula we have: 

7 = 3000 S,= 5.59 1b. 

F=4000 S,= 9.94 1b. 

7 = 5000 aSi = 15.53 lb. 

7 = 6000 >Si=22.36 1b. 

The values of S^^ for double belts will be nearly twice those 
given above. At a speed of 5000 ft. per minute the maximum 
tension per inch of width on a single belt designed by formula 
(137), if we call C = .5, will be: 

(30X2) +15. =75 lb. 

giving a factor of safety of eight or ten at the splices. 

In a similar manner we find the maximum tension per inch of 
width of a double belt to be: 

(50X2) +30 = 130 lb. 

and the margin of safety about the same as in single belting. 
A double belt is stiffer than a single one and should not be 



ROPE DRIVES 229 

used on pulleys less than 1 ft. in diameter. Triple belts can be 
used successfully on pulleys over 20 in. in diameter. 

119. Manila Rope Transmission. — Ropes are sometimes used 
instead of flat belts for transmitting power short distances. 
They possess the following advantages: they are cheaper than 
belts in first cost; they are flexible in every direction and can 
be carried around corners readily. They have, however, the 
disadvantage of being less efficient in transmission than leather 
belts and less durable; they are also somewhat difficult to splice 
or repair. 

There are two systems of rope driving in common use: the 
English and the American. In the former there are as many 
separate ropes as there are 
grooves in one pulley, each rope 
being an endless loop always 
running in one groove. 

In the American system one 
continuous rope is used passing 
back and forth from one groove 
to another and finally returning 
to the starting-point. 

The advantage of the English 
system consists in the fact that Fig. 99. 

one of the ropes may fail with- 
out causing a breakdown of the entire drive, there usually 
being two or three ropes in excess of the number actually neces- 
sary. On the other hand the American system has the ad- 
vantage of a uniform regulation of the tension on all the plies 
of rope. The guide pulley, which guides the last slack turn of 
rope back to the starting-point, is usually also a tension 
pulley and can be weighted to secure any desired tension. The 
English method is most used for heavy drives from engines 
to head shafts; the American for lighter work in distributing power 
to the different rooms of a factory. The grooves in the pulleys 
for manila or cotton ropes usually have their sides inclined at an 
angle of about 45 degrees, thus wedging the rope in the groove. 

The Walker groove has curved sides as shown in Fig. 99, the 
curvature increasing toward the bottom. As the rope wears and 




230 MACHINE DESIGN 

stretches it becomes smaller and sinks deeper in the groove; the 
sides of the groove being more oblique near the bottom, the older 
rope is not pinched so hard as the newer and this tends to throw 
more of the work on the latter. 

120. Strength of Manila Ropes. — The formulas for transmis- 
sion by ropes are similar to those for belts, the values for >S and 
¥ being changed. The ultimate tensile strength of manila and 
hemp rope is about 10,000 lb. per square inch. 

To insure durability and efficiency it has been found best in 
practice to use a large factor of safety. Prof. Forrest R. Jones 
in his book on Machine Design recommends a maximum tension 
of 200 d^ pounds where d is the diameter of rope in inches. This 
corresponds to a tensile stress of 255 lb. per square inch or a 
factor of safety of about 40. 

The value of /for manila on metal is about 0.12, but as the 
normal pressure between the two surfaces is increased by the 
wedge action of the rope in the groove we shall have the apparent 
value of/: 

/^=/-^sin ^ where • 

a = angle of groove. 
For a = 45°to30° 

/^ varies from 0.3 to 0.5 and these values should be used in for- 
mula (134). 

(l -e-f') in this formula, for an arc of contact of 150 degrees, 
becomes either .54 or .73 according as/^ is taken 0.3 or 0.5. 

If T^ is assumed as 250 lb. per square inch, the force R trans- 
mitted by the rope varies from 135 lb. to 185 lb. per square inch 
area of rope section. 

The following table gives the horse-power of manila ropes based 
on a maximum tension of 255 lb. per square inch. 



ROPE DRIVES 



231 



TABLE LVIII 

Table of the horse-power of transmission rope, reprinted from the trans- 
actions of the American Society of Mechanical Engineers, Vol. XII, page 
230, Article on " Rope Driving" by C. W. Hunt. 

The working strain is 800 lb. for a 2-in. diameter rope and is the same at 
all speeds, due allowance having been made for loss by centrifugal force. 



Diameter 
rope, 
inches 


Speed of the rope in feet per minute 


Smallest 

diameter 

pulleys, 

inches 


1,500 


2,000 


2,500 


3,000 


3.500 


4,000 


4,500 


5.000 


6,000 


7,000 


1 


3.3 


4.3 


5.2 


5.8 


6.7 


7.2 


7.7 


7.7 


7.1 


4.9 


30 


i 


4.5 


5.9 


7.0 


8.2 


9.1 


9.8 


10.8 


10.8 


9.3 


6.9 


36 


1 


5.8 


7.7 


9.2 


10.7 


11.9 


12.8 


13.6 


13.7 


12.5 


8.8 


42 


• u 


9.2 


12.1 


14.3 


16.8 


18.6 


20.0 


21.2 


21.4 


19.5 


13.8 


54 


1* 


13.1 


17.4 


20.7 


23.1 


26.8 


28.8 


30.6 


30.8 


28.2 


19.8 


60 


11 


18.0 


23.7 


28.2 


32.8 


36.4 


39.2 


41.5 


41.8 


37.4 


27.6 


72 


2 


23.1 


30.8 


36.8 


42.8 


47.6 


51.2 


54.4 


54.8 


50.0 


35.2 


84 



121. Cotton Rope Transmission. — Cotton rope is more expen- 
sive than manila in its first cost, but has a greater efficiency and 
a longer life than its rival. Instances are given where cotton 
ropes have been in continuous service for periods of fifteen, 
twenty-five and even thirty years. The rope of three strands 
without a core is most flexible and durable as there is good 
contact between the working strands and no waste room. 

Mr. Edward Kenyon gives the following values for the power 
which can be safely transmitted by good three-strand cotton 
ropes running on pulleys not less than thirty times their respec- 
tive diameters (English system).^ (See next page.) 

The horse-power at any other speed will be in proportion to 
the speed. It will also be noticed that the horse-power is 
proportional to the square of the diameter of the rope. Mr. 
Kenyon gives figures for the speed as high as 7000 ft. per minute, 
and reports actual installations where ropes are running success- 
fully at this speed. He makes no allowance for centrifugal 
force and denies that this has any appreciable effect on the 
driving power or the durability. 

Mr. Kenyon' s figures have reference only to ropes used in 
single plies as in the English system. 

» Am. Mach., July 8, 1909. 



232 



MACHINE DESIGN 



TABLE LIX 

Horse-power of Cotton Ropes — Velocity 1000 Ft. 

PER Minute 



Diameters in inches 


Horse-power 


Rope 


Smallest pulley 


1 

H 

n 

If 
li 
If 
If 

2 


30 
34 
38 
41 
45 
49 
53 
57 
60 


3.3 
4.1 
5.1 
6.1 
7.4 
8.6 

10. 

11.5 

13. 



He calls especial attention to the use of casing in high-speed 
pulleys to reduce the air resistance. 

122. Wire Rope Transmission. — Wire ropes have been used to 
transmit power where the distances were too great for belting or 
hemp rope transmission. The increased use 
of electrical transmission is gradually crowding 
out this latter form of rope driving. 

For comparatively short distances of from 
100 to 500 yd. wire rope still offers a cheap and 
simple means of carrying power. 

The pulleys or wheels are entirely different 
from those used with manila ropes. 

Fig. 100 shows a section of the rim of such a 
pulley. The rope does not touch the sides of 
the groove but rests on a shallow depression 
in a wooden, leather or rubber filling at the 
bottom. The high side flanges prevent the rope from leaving 
the pulley when swaying on account of the high speed. 

The pulleys must be large, usually about 100 times the diam- 




FiG. 100. 



ROPE DRIVES 



233 



eter of rope used, and run at comparatively high speeds. The 
ropes should not be less than 200 ft. long unless some form of 
tightening pulley is u^ed. Table LX is taken from Roebling. 
Long ropes should be supported by idle pulleys every 400 ft. 

TABLE LX 

Transmission of Power by Wire Rope 

Showing necessary size and speed of wheels and rope to obtain any desired 
amount of power. 



Diameter 

of wheel 
in feet 


Number 
of revolu- 
tions 


Diameter 
of rope 


Horse- 
power 


Diameter 

of wheel 

in feet 


Number 
of revolu- 
tions 


Diame- 
ter of 
rope 


Horse- 
power 


4 


80 


5-8 


3.3 


10 


80 


11-16 


58.4 




100 


5-8 


4.1 




100 


11-16 


73. 




120 


5-8 


5. 




120 


11-16 


87.6 




140 


5-8 


5.8 




140 


11-16 


102.2 


5 


80 


7-16 


6.9 


11 


80 


11-16 


75.5 




100 


7-16 


8.6 




100 


11-16 


94.4 




120 


7-16 


10.3 




120 


11-16 


113.3 




140 


7-16 


12.1 




140 


11-16 


132.1 


6 


80 


1-2 


10.7 


12 


80 


3-4 


99.3 




100 


1-2 


13.4 




100 


3-4 


124.1 




120 


1-2 


16.1 




120 


3-4 


148.9 




140 


1-2 


18.7 




140 


3-4 


173.7 


7 


80 


9-16 


16.9 


13 


80 


3-4 


122.6 




100 


9-16 


21.1 




100 


3-4 


153.2 




120 


9-16 


25.3 




120 


3-4 


183.9 


8 


80 


5-8 


22. 


14 


80 


7-8 


148. 




100 


5-8 


27.5 




100 


7-8 


185. 




120 


5-8 


33.0 




120 


7-8 


222. 


9 


80 


5-8 


41.5 


15 


80 


7-8 


217. 




100 


5-8 


51.9 




100 


7-8 


259. 




120 


5-8 


62.2 




120 


7-8 


300. 



PROBLEMS 

1. Design a main driving belt to transmit 200 horse-power from a belt 
wheel 18 ft. in diameter and making 80 revolutions per minute. The belt 
to be double leather without rivets. 

2. Investigate driving belt on an engine and calculate the horse-power 
it is capable of transmitting economically. 



234 MACHINE DESIGN 

3. Calculate the total maximum tension per inch of width due to load and 
to centrifugal force of the driving belt on the motor used for driving machine 
shop, assuming the maximum load to be 10 horse-power. 

4. Design a manila rope drive, English system, to transmit 400 horse- 
power, the wheel on the engine being 20 ft. in diameter and making 60 revo- 
lutions per minute. Use Hunt's table and then check by calculating the 
centrifugal tension and the total maximum tension on each rope. Assume 

5= — where v= feet per second. 
oO 

5. Design a wire rope transmission to carry 150 horse-power a distance of 
one-quarter mile using two ropes. Determine working and maximum ten- 
sion on rope, length of rope, diameter and speed of pulleys and number of 
supporting pulleys. 

REFERENCES 

Manufacture of Belting. Power, Feb., 1903. 

Steel Belts. Am. Mach., Jan. 24, 1908; Eng. News, Oct. 14, 1909. 

Belts vs. Ropes. Power, Dec. 1, 1908. 

Lewis' Experiments on Belts. Tr. A. S. M. E., Vol. VII, p. 549. 

Transmission of Power by Belting. Tr. A. S. M. E., Vol. XX, p. 466; 

Vol. XXXI, p. 29. 
Various Systems of Rope Transmission. Am. Mach., July 8, 1909. 
Rope Driving. (Hunt.) Tr. A. S. M. E., Vol. XII, p. 230. 
Working Load for Ropes. Tr. A. S. M. E., Vol. XXIII, p. 125. 



CHAPTER XIII 

DESIGN OF TOGGLE-JOINT PRESS 

123. Introductory Statement. — In discussing the subject of 
Machine Design much time may be saved by assuming some simple 
machine and illustrating methods in design by a fairly complete 
analysis of all the important theoretical calculations. Such a 
layout at once gives the scope of the work and protects the beginner 
from so much "working in the dark." An assignment may then 
be made, differing in a lesser or greater degree from the illustrated 
design and a complete analysis required of all parts of the machine. 
After the student's experience with the first design he will need 
the second one developed less elaborately and possibly the third 
one not at all. 

Design No. 1 is meant especially to cover static forces, i.e.^ 
simple applications of members in tension, compression, flexure 
and shear. A good illustration of this is the toggle-joint press. 
Machines of this class are sometimes used in forming thin sheets of 
copper and brass into articles for ornamental purposes, conse- 
quently it is a useful tool. Plates C-1, C-2, and C-3 show a 
design of a small machine and. are inserted to give an idea as to the 
arrangement of the drawings. The design was worked up on 
three 12 in. X 18 in. sheets; two of details and one assembly view. 

It is urged that the designer regard these sheets merely as 
illustrative of a good drawing room job and that, from the stand- 
point of ideas, he will cultivate originality and make a design as 
nearly independent as possible. 

Alternative Designs will be found at the end of this chapter. 
These may be substituted for the regular designs if preferred. 

In designing a complete machine each part should be worked 
up as an independent unit but with all available information as 
to its relation to the other parts composing the machine. Be- 
fore attempting to develop any individual part, the designer should 
have a good idea of what the machine looks like. Free-hand 

235 



236 MACHINE DESIGN 

sketching should be insisted upon. These sketches when satis- 
factory should become a part of the report and be handed in with 
the finished drawings. Calculate by rational formula every part 
that will admit of such treatment. Where the conditions of 
stresses are not well known apply empirical rules and the best 
approximations possible. In any case the judgment of the 
designer must be used to modify and check even the best rational 
or empirical rules. Theoretical deductions should not be mini- 
mized but good j udgment should be emphasized. All calculations 
should he saved until the entire design is finished and these should 
be kept in the exact order of development. Sometimes a part that 
is at first considered wrong may later be found to be correct and 
recalculation is avoided. Occasionally it is necessary to review 
part of the calculations to prove some part of the design. Where 
the theoretical work is neatly made and logically arranged this 
may be done without much loss of time. In the analysis of the 
forces and the calculations of the various parts a high degree 
of refinement should be aimed at for the sake of showing how prin- 
ciples are applied, even though the illustrative piece may not 
demand a very thorough analysis. The object sought is not so 
much that a machine be designed by the student as that he 
be fortified with the ability to analyze a problem and that he be 
able to apply to it the correct principles of design. 

124. Drawings. — The following dimensions are suggested for 
the cutting sizes of the sheets. The designer is at liberty to make 
his own selection from these sizes. It is suggested, however, 
that the sheets be taken as small as will admit of a clear and 
distinct set of drawings. 

24 in. X 36 in.— Size A 

18 in.X24 in.— Size B 

12 in.XlS in.— Size C 

9 in.Xl2 in.— Size D 

Scale. — Any scale may be taken which will show clearly all the 
details and give a good arrangement on the sheet. Details may 
have different scales on the same sheet if so desired. When 
this is done each detail should have the scale given. 



PLATE C 1 



50"- 



ni 




S 



/^•^ 



^S^ 



58" 








Note:|See details on drawings 
C— 2 land 0—3. 



<> 



y 






^ 



^ 



TDGBLB JOINT PRESS 
ASSEMBLY 

Scale — 4-*'^^' 
F^JTdue UoiV/a TSitV L oFoyetfCr Ii7cl 

■ Clp^okcd by 

Clip proved 






INSTRUCTIONS CONCERNING DRAWINGS 237 

Border Line. — A margin of J in. should be left between the 
border line and the edge of the finished sheet on the top, bottom 
and right end, and J in. on the left end to allow for punching and 
fastening. 

Name Plate. — Make the name plate or title at the lower right- 
hand corner to cover a space about 2\ in. X3^ in. If any other 
standard corner is preferred other dimensions may be sub- 
stituted. No border line need be drawn around the name plate. 
It would be well for each designer to make a standard corner 
plate to be used below the various tracings when working up this 
part. 

All drawings will be carefully worked up in pencil and turned 
in to the instructor. The instructor will give them to another 
designer who will be responsible for the checking. Checking 
will be done in the form of notes on a separate paper and attached 
to the drawings. These notes and drawings will then be returned 
to the designer for approval and corrections. When the designer 
traces his drawings, or such part of them as may be selected by 
the instructor, he will obtain the signature of the checker to 
them and submit the same with the checker's notes to the 
instructor for approval. 

Each designer should have experience not only in planning and 
executing well his own designs, but he should take up designs of 
other men and offer suggestions and criticisms upon their work. 
One way to obtain this experience has been suggested above. 

In checking up the work of another man the following points 
should be observed: 

(1) General appearance of the design relative to workman- 
ship and execution, arrangement of drawings, notes, dimen- 
sions, etc. 

(2) General design relative to proportion, strength and 
arrangement of parts. This is to be merely the checker's im- 
pression and need not require the checking of the original 
calculations. 

No drawing should be retained longer than one exercise and 
at the completion of the checking should be returned to the 
designer. It is estimated that any set of drawings may be 
checked in this way within two hours' time. No notes or marks 
will be made on the drawings but special paper will be provided 




238 MACHINE DESIGN 

for this purpose. In looking over the drawings finally, the in- 
structor will give credit to the work of the checker as well as to 
that of the designer. 

In all this work some standard text on mechanical drawing 
should be adopted as reference concerning arrangement of views, 
sectioning, cross-hatching, lettering, and the like. 

Every dimension should be clearly shown so that no measure- 
ments need be taken by scale from the drawing. 

All dimensions should be given in round vertical figures, 
heavy enough to print well. No diagonal-barred fractions, thin 
or doubtful figures should be accepted. 

All dimensions should be given 
in inches. 

All dimension lines should be 
made as light as will insure good 
printing and should have a central 
space for figures. 

All dimensions should read in 
the direction of the arrows. 
Avoid crowding the dimensions to the center of any detail. A 
much better way is by the use of projected lines as shown by 
Fig. 101. 

All detailed pieces should be accompanied by a shop note 
or call &s "C. I. One wanted'^ '^M. S. Two wanted''; "Finish 
all over;" "Turned for a shrinking fit;" etc., etc. 

The following abbreviations will be considered satisfactory in 
these calls: 

C. S Cast steel. f Finish (see sheets of details). 

C. I Cast iron. B. b. t.. Babbitt metal. 

W. I Wrought iron. D Diameter. 

M. S Machine steel. R Radius. 

125. Calculations. — Each designer is expected to draw up a 
report in parallel with the design. This report should contain 
such free-hand sketches as relate to the calculations, also a full 
report of the calculated sizes and accepted sizes of the different 
parts of the design, and be submitted in a manila cover with the 
finished tracings and drawings. 



PLATE C 2 



•6" — — »!♦- 



3S 



fl 



F^tl 



■4- hole,s 



2.TQp,rdeep, foT- :T 




5o"- 




Slidif^a Die Head. G-i. 
Oije -reoel 

Pii^ i**>c 2 g W.I Ofje Teq'd. 
Dt'iva 17 opd Pile riu»n 



Drill %" For Pi9 ^"a 8" M5 
Of^e -re«|,'d 



a* 



[ Drin ,%'*oijd C.S 
^ FoT l'xl^*"npocksc. 
ef«<^^d 



1 I I r^^-^ 




e" 



^ 



tf 



WJ 






DSe Head Ci. 



i Teo'd 



:or g'x^-'bolt. 



£4 C^, 



.<^. 



id. 



oi^d C<S. Pot 
npoc^.sc 2. Teoci. 




B«d C. t. 
Orj© Treod. 




Grip Plot* C.l, 

O7C Tfeo'd. 






Dt.IIITFot Pii? 







i^-^i*- 



r^4-^ 



$;i 




8'- 



H 



^^^ 



— r: 






4u 



L— eveTOTor^d CI. 
Oqo req,'d. 

Nota:- 

drowiiyo''*' C-J9I. 



TD5GLE JDINT PRESS 
DETAILS 

Scal©.ie*'Td 8 size 
r^rdue UipiversiTy LoFoyette, l^d 

:C flecked by 
\o ppro\/ed 



TOGGLE-JOINT DETAILS 



239 



Design No. 1 

Toggle-joint Press 

126. Analysis of the Forces Involved. — Referring to Plate C-1, 
it will be seen that the acting forces can be represented in the 
following force diagram, with the direction of the forces repre- 
sented by arrows. 




Fig. 102. 

Each designer will be given a value for W, I, V and (f). In all 
the designs may be taken at 10°, assuming that the maximum 
load will be carried at this position and that the lever arm will 
then be horizontal. 

In the assignments for a number of designs the range of values 
may be as follows: 

W = 2m, 300, 400 1000 lb. 

Z = 4, 4.5, 5, 5.5 10 ft. 

Z' =for large sizes, 6, 8, 10 12 in. 

for small sizes 6, 6.5, 7 8 in. 

Selecting for our analysis the following values: "pr = 100 lb.; 
Z = 5 ft. 3 in.; V = 7 in.; and ^ = 10 degrees, we have from the 
force diagram 

Wl 100X63 



W = 



W" = 



V 7 

WQ,-V) 100X56 



900 lb. 



= 800 lb. 



W 



900 



= 2591.4 lb. 



V 

^^' ^^'2 sin. 96 .3473 
TF3 =W, cos 9^ = 2591.4 X. 98481 =2552 lb. 
900 



W 



= 450 lb. 



240 



MACHINE DESIGN 



127. Lever. — The formula for calculating beams in flexure, 

Art. 4, is M = SZ, where M= bending moment in pounds- 
inches, S = working fiber stress in pounds and Z = resistance of 
the section or modulus. In any section of the lever transversely 
across the axis let h and h be the breadth and the height of the 
section respectively. The designer must here decide if the beam 
is to have parallel sides, in which case b will be constant for all 
sections, or taper sides, in which case a certain ratio of 6 to /t 
would be used. The best' way to decide which to use is to find 
the size of the sections at two critical points as g and c, Fig. 
103 (c is the fulcrum and g is any point near the handle), for 
each case and select between them. Assuming ;S = 8000 for 



d 



2 ijit 



-22- 



■22- 



n 



Fig. 103. 



wrought iron or mild steel, h = l, and disregarding the hole at c, 
which has little effect since the fiber stress of any section ap- 
proaches zero at the center, our formula M = SZ gives 
(section at ^) 100 X Q = SOOOXlXh^-^Q; h= .67 in. 
(section at c) 100X56 = 8000x1 X/i'-^ 6; /i = 2.05 in. 
This beam would have a better shape and would also be lighter 
if the thickness be reduced below 1 in., say to .75 in. With this 
value the formula gives 

(At ^) lOOX 6 = 8000X.75X/i'^6; /i= .77 in. 
(At c) 100x56 = 8000X.75X/i'-^6; /i = 2.37in. 
These values give a well-shaped beam, having a section .75 in. 
X.77 in. at g and .75 in. X 2.37 in. at c. 

On the other hand, suppose a ratio of 6 to /i = J, to be desired, 
the problem becomes 

(At g) 100X6 = 8000 h^^24:; h = 1.22 in. 
and 6 = 1.22 --4 = .3 in. 



PLATE C 3 





Drill ^** To^ P;.p^"x2.8"w» 0<pe rec^d. 

■yDlrivo ip 09c/ Fi/e pJuaK. 



D-rill i" Tor 



Pi? ^VS 



cy 







« 



(0 



•■4 



Note:- 

See o«senpbly 
oij (J'w'a C'I9I. 



Co77ectiipQ Li7k. Y'l 

Also Tec^'ci 070 ^ bo/t, 5 8 »®7*)i 

riQiikftd oil over , threaded t with two 
lo<sk-i}btts . PocAct off to a tfiicl(7e&& oK 



Dt-.IIJ" 




TDGGLE JOINT PRESS 
DETAILS 

^cale, 



^ size. 



TOGGLE-JOINT DETAILS 241 

(At c) 100x56 = 8000/1^-^24; /i = 2.56 in. 
and 6 = 2.56 4-4 = .64 in. 
section at 5^= .3 in. X 1.22 in. 
section at c = .64 in. X 2. 56 in. 

The above gives the method of determining the size of the 
section at any point of the beam. Sections should be taken at 
regular intervals of length and a diagram plotted from the results. 
One section only need be taken between a and c, say at o midway 
between. This diagram when completed will show the beam to 
take the form of a curve similar to Fig. 104. It may be found 
convenient, however, to approximate this curve with a straight 
line as x y. This would be satisfactory for strength and would 
be more easily constructed. 

It will be noticed that the bending moment becomes zero at the 
points a and p where the loads are applied. This would theoret- 
ically give no size to the handles and make it impossible of 




Fig. 104. 

construction. Some satisfactory design of handle or hub must 
be made at these points with sufficient size to carry the pins or 
bolts, each hub to have the sides and edges of the beam filleted 
into it in a neat manner. See Plate C-3. A handle can be 
placed at p for all loads of 300 lb. or less and a drilled hub for 
larger loads so that a small air or steam cylinder can be attached. 
A similar hub will be added at a, for connection to the post at 
the rear. 



128. — The following shapes may be found useful in designing 
the lever. 

Shapes at p. — The size and shape of the handle or hub at this 
end will be largely a question of neatness, since the load carried is 
very small. The pin, if one is used, may be calculated for 



242 



MACHINE DESIGN 





{ 



X 



Fig. 105. 



f 





Ftg. 106. 



TOGGLE-JOINT DETAILS 



243 




Fig. 107. 



double shear to get the minimum size allowable, but this size 
will probably be so small that it will be necessary to increase the 
size of both pin and hub to add symmetry to the design. Such 
points as this call for special investigation. Any piece of a 
machine may be made extra strong, if necessary to harmonize 
with the other parts of the machine, but the reverse is not the case. 

Construction of the Joint at a. — Referring to Fig. 106, shapes 
A and B would be preferred. In most cases the standard would 
be made of cast iron and could easily be cored out to fit over 
the lever arm end rather than to fit the arm end over the stand- 
ard as at C. The only calcu- 
lations necessary for this end 
of the lever, besides figuring 
the pin, are those that deter- 
mine the diameter of the hub 
and the length of the hub. 
It is reasonable to assume 
that the diameter of this hub 
should be made equal to the 
diameter of the cast hub of 
the standard. To illustrate: at a, a tensional force of TF" is 
acting upward and this force is resisted by four cast iron areas 
on the section, RS, equal in total area to R'S\ of the standard 
(Fig. 107). 

These four areas are produced by passing a plane through the 
standard along the line RS. Each area is equal to hh and should 
be figured for cast iron in direct tension by the formula W = SA. 
In making this calculation the ratio of h to h may be assumed. 
Having figured the pin for double shear by the formula W^^ = 2SA. 
find the diameter of the pin and add to it 2h, which will give the 
diameter of the cast hub and consequently the diameter of the 
lever end. If S for shear in wrought iron be taken at 5000 lb. 
per square inch, the diameter of the pin will be .33 in. or, say | in. 
If S for tension in cast iron be taken at 1500 lb. per square inch, 
the area hh will be .133 square inch, from which, if b be taken at 
i in., h becomes .53 in. This would make the diameter of the 
hubs at a, If in. 

It will be next in order to find the length of the hub at the lever 
end, also the corresponding values of the standard top. These 



244 MACHINE DESIGN 

are determined largely from the crushing strength of the pin. 
First examine h of the standard to see if the assumed I in. is 
sufficient. The part of the pin in the casting and the part in the 
lever are both subjected to a crushing force. The resistance of 
the pin to crushing is in proportion to the projected area of that 
part of the pin involved. 

In Fig. 108 let the pin be cut by a horizontal plane through its 

diameter 1, 6, 7, 4, corresponding to the plane along RS of the 

standard. 1, 2, 3, 4 and 5, 6, 7, 8 are the projections of the parts 




Fig. 108. 

included within the arms of the standard and 2, 5, 8, 3 is the pro- 
jection of that part included within the lever end. The diameter 
of the pin has previously been figured to resist shearing along 
the two planes 2, 3 and 5, 8. Now it is necessary to find the 
length 1, 2 and 5, 6 such that these parts will be safe from crush- 
ing. For the part in the casting, 2hd = 2XiXi = TQ sq. in. = 
areas 1, 2, 3, 4 + 5, 6, 7, 8. If now the factor of safety for the 
wroughtiron pin be taken so that 5000 is a safe value for shear, 
Ss, the pin will sustain W = SsA = ^\X5000 = QSS lb. safely. 
This we find is greater than the load TF'' actually pulling on the 
standard so that part of the pin within the castiron standard is 
safe. If it had been found that 2bd was so small that the load 
it was capable of sustaining safely before crushing was less than 
the load applied, then either & or d or both would be increased. 
If d were increased without changing b then the hub diameter 
would be increased this amount above the calculated size of If in., 
but if b were increased, the areas bh would be stronger than the 
calculated value and h could be reduced accordingly, if it were 
considered necessary. 



TOGGLE-JOINT DETAILS 



245 



By the same line of reasoning the length of the pin within the 
lever would determine the mimimum length of the lever hub to 
resist crushing. This would be 2h = ^ in. From inspection it is 
readily seen that the thickness at a must be necessarily increased 
to that of the lever section. This at c is .64 in. 

In every fastening of 
this kind, investigation 
may be made for shearing 
of the pin, the strength of 
the sections around the 
pin, and the crushing of 
the pin, within both lever 
and standard. 





129. — In calculating the 
size of the section at c the 
hole was not considered. 
The error introduced by 
this is very slight and in 
most cases may be neg- 
lected. The fiber stress 
in the cross-section of the 
arm varies from zero near 
the center to a maximum 
at the edge as shown in 
diagram B, Fig. 109, where 
by proportion we can 
readily obtain the relative 
resistance offered by the 
metal at the center as com- 
pared to that at the edge of 
the section. The loss at 
the center is more than 
taken up by the addition 
of a fraction of an inch at the edge or a very small boss around 
the hole. If the hole in any case should be large, a modulus 
could be selected for this hollow section, and the exact sizes 
obtained. 

The pin would be calculated in double shear — as at a. . 



T 



Fig. 109. 



246 



MACHINE DESIGN 



The size of the boss, if any be added, is largely optional and is 
put on for finish. 




130. Screw Fastening for Standard. — In deciding upon the 
kind of fastening between the standard and the bed, it would be 
well to first examine it regarding the turning moments about a, 
Fig. 110, where W''b + W^h,-W\y = WJ' + WyV\ Assume 
b = b' = 3 in., h, = 2m., V = 5m. and r = 1 in. then with Tr" = 800, 
1^3 = 2552, and W\ = 4.50 lb. We have 5 Wa^ + Wy = Q154: inch- 
pounds. 

If Wx = Wy then 6 TF^, = 6154 or W ;, = 1026 lb. This is equiva- 
lent to a ^-in. bolt. Suppose W y, 
because of its location, to be of little 
value in resisting turning about a, 
then 5 TF^ = 6154 and TF:, = 1231 lb. 
= approximately iVii^- bolt. If 
more than one bolt is used along the 
line W X or W y then the total bolt 
area at the root of the threads may 
be the equivalent of that given 
above. 

Next examine the joint for a sum- 
mation of all vertical forces. 

W" —W\ = force holding stand- 
ard to bed = TF:. + Wy. If Wx = Wy 
then, 2 TF^ = 800-450=350 lb. and 
Fig. 110. 17:, = 175 lb. 

Since this force is less than that 
obtained by moments it need not be considered. 

Next examine the joint for a summation of the horizontal 
forces. In this the force W^ tends to shear the bolts off in a 
plane with the top of the bed. It also acts upon the flanges to 
shear the casting inside the bolt holes. Considering the bolts 
first 

W^ = SA. If we t ake >S = 5000, then 
2552 = 5000A; A = .51 sq. in. of bolt area. 

If the bolt shears at the root of the thread, as would be the case 
with a cap screw, we need at least four J-in. cap screws. 



TOGGLE-JOINT DETAILS 



247 



In the second case, if the flange is, say 6 in. long and t in. 
thick, we have for the two sides 2552-2x6 tS. Let aS = 1500 
for cast iron and find i = approximately .15 in. 

This would, of course, be made thicker, say i to J in., for the 
appearance and good proportion of the casting. 

In the above discussion of the standard fastening, the part 
most liable to fail would be the shearing of the bolts. This 
might not be true in every case; for example, if h^ were very great 
when compared to V, the failure of the joint would probably be 
by moments about a. The above calculations would be modified, 
also by the arrangement of the bolts or cap screws. 

It is well in every case to examine a joint from all standpoints 
and design for the greatest requirement. 

131. Standard. — The design of the standard would depend 
largely upon the magnitude of the force to be resisted. In the 
smaller machines it would undoubtedly be made of cast iron and 






Fig. 111. 

as such the upper end would be as shown in the preceding 
paragraph. In the larger machines the standard would be 
made of wrought iron or steel plates, in which case the sizes of 
the standard and lever end would be calculated from different 
values of S than those used for cast iron. fj|rj ■ ^ -: 

The cross-section of the body of a cast iron standard may be 
shaped as in Fig. 111. Assuming the areas to be equal, the 
strongest section to resist any bending action that may come 
upon it, is D. 

The lower end of the standard would be planned to receive the 
rod TFg, and would have a flange for fastening to the top of the 
bed. Fig. 112 shows some of the shapes that may be used. 

The pin at the base is figured for double shear by the formula 
W2 = 2SA. 



248 



MACHINE DESIGN 



Note. — When the constant 2 is used in the formula for double 
shear the result is the single cross-section of the piece. When 
this constant is omitted as in TF = SA the result is the combined 
cross-sectional area. 




Fig. 112. 

132. Toggle. — There are three ways in which the toggle may 
fail at the central joint: by shearing the pin, by bending the pin 
and by crushing the pin. In Fig. 113, J5 shows a very simple 
arrangement of this point. To obtain the size of the pin in this 
case to resist shearing 



<_^m 




IVi 

2 



m 



c 



B II 



w' 



Wi 



< 



c 



! 2 



2 



2 



Fig. 113. 



Wi = W2 = 2SA. If aS = 5000, then 
2591.4 = 2X5000A.; A = .26 sq. in. and d = ,5S say f in. 
It is readily seen that the pin would be found to be the same 

. . W 

size if the load ~ were figured for single shear as if W^ were 

figured for double shear. 

To obtain the size of the pin to resist bending assume some 



TOGGLE-JOINT DETAILS 



249 



length of pin between the outer forces 
solve by the formula W'l-^S = SZ. 



as 2 in. or 3 in., and 
4. There might 



El 

2 
See Art 

be a question raised here concerning the proper formula to use 
for the bending moment, i.e., fixed ends or free ends. With the 
two ends of the pin held somewhat rigidly between the two sets 
of resisting forces, it is in about the same condition as a beam 




Fig. 114. 



If >S = 8000 and 

11 



.65 say -^-^ m. 



fixed at the ends and loaded at the middle 
1 = 2, then 900x2-^8 = 8000X7^(^'-^32; d 

The toggle action on the pin at the center requires that the 
smaller force W should come at the center of the length of the 
pin as shown in A and B, Fig. 113. If the heavier force W^ or TFj 
acts at the center of the pin it would cause an unnecessary bend- 
ing as shown in C, and would require too large a pin to resist 
this stress. 

Fig. 114 shows other methods of designing the toggle. 

Concerning the crushing of the pin see Art. 128. 



250 



MACHINE DESIGN 



133. Fig. 115 gives some shapes of toggle members. A, B, C, 
and D are usual shapes of the horizontal members. A and B 
have split ends and are necessarily hard to forge and machine. 
C is the simplest form. This form is sometimes modified by 
adding bosses to one or both sides as shown in D. The vertical 
member may be constructed solid as at E or adjustable as at F. 




I 




Fig. 115. 



134. Die Heads. — The sliding head receives the thrust W^ 
from the toggle and moves along the top of the bed toward or 
from the stationary head. It must be a good fit to the bed top 
having a free sliding contact but no side motion. The stationary 
head must be planned for longitudinal adjustment and for fasten- 



TOGGLE-JOINT DETAILS 



251 



ing rigidly to the bed top when desired. Suggestions for attach- 
ing these heads are shown in Fig. 116. Rectangular and V- 
shaped ways are used, some having adjusting gibs and some 
plain. A is the simplest form and may be grooved from the solid 
or held down by plates. In such a design the overlap below the 
top of the bed should be made sufficiently strong to resist the 
turning action from W^- B and C show the application of gibs 
between the sliding head and the bed to take up side slack. In 
some classes of machines gib arrangements are essential. If, 




Fig. 116. 



however, heavy side thrusts are involved the form C is question- 
able unless made very heavy and strong. With the bed planed 
to an angle as at C and D, the latter would be considered the 
stronger. 

Sliding Head. — Since the sliding head cannot be rigidly fastened 
to the bed, it must be fitted to a set of guides. The most common 
fastening is shown in Fig. 117. Having the forces W^ and W^ 
(resultant forces from TFJ acting on the pin and allowing all the 
reaction from the die to fall at the upper point of the head, say 
4 in. above the bed, we have a cantilever beam projecting upward 
from the bed top and acted upon by three forces tending to 
break the beam at some section as ao. Any leverages may be 



252 



MACHINE DESIGN 



selected other than 4 and 2 but these are given for the sake of 
argument, the actual values used depending largely upon the 
kind of die used between the two heads. ^ It is evident that the 
two forces opposing each other (W^, action and reaction) will 
have the same value. These moments will cause stresses in any 
section under investigation. Suppose the line ao to be the 
weakest section in the beam. The tendency to break here is 
resisted by two metal sections, each h inches in width and h inches 
in height, or, by one section 2 b inches by h inches. The fiber 
stress caused by the moments from the two TFg forces will cause 
a maximum tension at o which becomes less as it approaches a. 
This tensional fiber stress at o will be partially neutralized by a 
downward force TF4 distributed more or less uniformly over the 
area 2 bh; the final stress at being the algebraic sum of the 
two. Let Sp= pressure per square inch acting perpendicularly 





Fig. 117. 



to the bed top over section 2 bh, Sm = tensional fiber stress at 
due to the moments and St = resulting fiber stress at 0. 

Taking W^ in two moments about ao and W^ in direct pressure 
we have W^ = Sp A and M = Sm Z, from which we obtain Sp = 



450 ^ , ^. , , ^ 2552X2X6 15312 
r^ 1 1 due to direct pressure and Om= oT^^ ^~hh^ 

due to the summation of the moments. Now if Sm—Sp=St; 
also ii h==5 in. and *Sf = 1500 we have 6 = approx. | in. 

If the fiber strength of tension and shear in cast iron be taken 
the same, then b' = b approximately. 

In like manner the reaction TF3 from the die may be taken at 
the bottom instead of the top of the sliding head, and the turning 
moment figured in this way to see if there is greater danger to 
the section than when taken at the top. 



TOGGLE-JOINT DETAILS 



253 



Other investigations may be made for this fastening. If the 
projection 5' were fastened on with screws the calculations would 
be worked up in a similar way to the fastening at the base of the 
standard. 

Stationary Head. — As in the sliding head, it is assumed that the 
stationary head is properly desig- 
ned above the bed top and that the 
fastening only is in question. 
Fastenings for small machines will 
not be difficult but those for large 
machines will call for extreme care. 
The simplest fastening is shown 
in Fig. 118 and acts as a frictional 
resistance only. If Wt = tension on 
the bolt in pounds, W2 = 2552 lb., 

2/ = 4 in., and a; = 6 in., we have by moments, disregarding the 
benefit obtained from the overlap of the block around the 
frame, 2552x4 = 6TFi, or, Wt = 1702 lb. This will hold the 
block to the bed. It is now necessary to determine if the 
block will slip with this force binding the frame between these 



8 5S2* 




Fig. 118. 




Fig. 119. 



two friction surfaces. Let the coefficient of friction be- 
tween the block and the frame also between the washer and 
the frame be ^ == say .3, then the resistance due to friction is, 
by formula, 2(j)Wt = F and when applied to our problem is 
1021.2 lb. That is, with the conditions as stated, if W^ were 



254 



MACHINE DESIGN 




2532 



only 40 per cent as large as it now is the block would just slip. 
Since the bolt as figured from moments proves to be too small to 
keep the block from slipping, let us reverse the process and find 
how large a bolt will be necessary to hold the block against the 
force TFg. By substituting as above we have 2 X .3 X TF^ = 2552, 
from which TFi = 4253.5 lb. This force is being exerted at the 
root of the thread tending to elongate the bolt. With /S = 8000, 
this will give slightly greater area than .5 sq. in. and will 
require a bolt of approximately 1 in. diameter. It is evident 
from this that more than one bolt should be used, or that some 
other arrangement be substituted for the friction surfaces. In 
Fig. 119, A is very similar to Fig. 118, excepting that the lower 

surface is notched to protect it 
4 from slipping. The upper block 
may slip slightly, but this will 
cause a greater grip and a conse- 
quent increase of frictional resis- 
tance. A possible improvement 
on this, if the construction of the 
machine would permit it, would 
be to have the bolt at an angle as 
shown in the dotted lines. 
Let this angle be, say 30 degrees with the horizontal, then from 
Fig. 120, A, .3T sin a = resistance due to friction, and T cos 
a = horizontal component of the bolt tension. Combining we 
have, .866^ + . 3 X. 5^ = 2552, or, 

r=: 2512 lb. 

This will require a j^-in. bolt. 

Fig. 120, B, will cause a tension on the bolt (disregarding 
friction) of T'=2552 tan (/?-^2). Let /5 = 90 degrees then 
T" = 2552 lb., requiring a if-in. bolt. It is very evident that if 
friction were included in this it would reduce the bolt size 
somewhat. 

Let the student investigate this with friction included. 

C, Fig. 119 is probably not as strong in the shape of the tooth 
as A and B, but with a large tooth area the unit shear becomes 
small enough so that the teeth are not endangered. The vertical 
faces on the teeth reduce the vertical thrust on the bolt to a 



/ 1 


\ Z ^ 


XA-zs-- 


. 25S2 


/ 


/ 


Fig. 120. 



TOGGLE-JOINT DETAILS 



255 



minimum and permit the use of a bolt just sufficiently strong to 
protect the block from turning as in Fig. 118. 

D is arranged to have pins to fasten into the frame either 
through the block, or behind it. These pins keep the block 
from sliding and are calculated for shear, while the bolt is cal- 
culated to resist turning as in Fig. 118. 

Another way in which these fastenings may fail is by shearing 
the bolt. Assume W^ Fig. 118 entirely acting to shear, we have 

2552 
S= say 5000 ""'^^ ^^' ^^' ^^ ^^^^ ^^^^- ^^ ^^^^ ^^ taken as the 
full area of the bolt it would be -}|-in. diameter. This shows a 
requirement about equal to that for tension. In any form of 
fastening it is well to investigate both tension and shear and 
take the larger requirement. 

It should be understood that, if the block clamps over the 
edges of the frame on planed ways, this will assist the bolt in 
holding the block down and a smaller bolt may be used. 

Let the student investigate this as in the case of the sliding 
head. 

135. Frame or Bed.— The calculations for the frame will be 
found somewhat more complicated. Assume a simple type, say 




Fig. 121. 

of the same general shape and cross-section as Fig. 121. Assume 
also the force W^ acting at some point along the block face, say 
at the middle of the block, a distance of 2 in. above the top of 
the frame. Any other height may be taken but in all probability 
if the dies should not be parallel and they should strike hard at 
the top, this inequality would be accounted for by a slight 



256 MACHINE DESIGN 

springing of the bed. It may be assumed that- the force W^ will 
act somewhere near the center of the die before it reaches such 
a magnitude as to endanger the frame. This force tends to 
break the bed along some line as rs, and produces combined ten- 
sional and compressional stresses in the fibers of the section. 
Considering the part to the right of the section as free we have, 
Fig. 122, the fibers on the upper or weak side subjected to two 
tensional stresses, the sum of which should not exceed the safe 
fiber stress of the metal, i.e., S^+S2 = St; and the fibers on the 
lower side, subjected to a tensional and a compressional stress, 
the algebraic sum of which should not exceed the safe com- 
pressional fiber stress of the metal, i.e., Sj^ + { — S2) =Sc where 

Si = uniform tensional stress 

/S2 = stress due to bending 

St and Sc = combined stresses. 

To obtain iS^ and S2 on the tension side use the formulas W^ = 
S^A and M = S^Z and obtain 

W 

-^ = S^ where A = area of section in square inches and 

W (h-i-2 + 2) 

— - — ^ - = S2 where Z = modulus of section. It will be seen 

that the moment arm is the perpendicular distance between 

h 
the force and the center of the section. The value ^ would be 

changed for any other than a uniform section. See Art. 144. 

Having selected the section of the bed as Fig. 121, we find the 
modulus to be 

Z = ^r-. X2 See Art. 4. 

6 h 

It will be necessary here to select some values for 6, V , h and h' 
and make a trial solution. Take 6=2 in.; 6^ = 1^ in.; /i = 6 in. 
and /i'=4 in. With these values we find A = 12 sq. in. and 

Si= ..^ =212.7 lb. per square inch 

also Z = 18.8 and 

^ 2552 (3 + 2) _^_ „ . , 

S^ = -^-- — ^ = 679 lb. per square mch. 

/Sf = ASi + >S2 = 679+212.7 = 891.7 lb. per square inch. 



TOGGLE-JOINT DETAILS 



257 



Since the usual value of St for cast iron is 1500 to 2000, this 
shape and size of section would be stronger than necessary. 

Now, if the figures of the section be changed to read b=2 in.; 
6' = liin.; h = 5 in.; and h' =4 in. the value becomes 



^^1 = 



2552 



319 lb. per square inch, and S2 = 



2552 (21 + 2) 



= 1126 lb. per square inch. 

S^+S2 = St = S19 + 1126 = 14:4:5 lb. per square inch. 




Fig. 122. 

This seems to agree very well with the safe value of cast iron in 
tension, and may be used. Since this is a symmetrical section and 
since cast iron is much weaker in tension than in compression, 
the latter will not need to be investigated and the above figures 
can be accepted for the sizes of the bed. With a section that is 
unsymmetrical it would be necessary to investigate both sides 
of the section. See Art. 144. 

Having found the shape of the simple section it is possible to 
modify it to a certain degree without affecting the calculations 
seriously. To illustrate, the portion abed, Fig. 123, may be 
lopped off and added to the inner side at a'Vc'd' without affecting 
the modulus. Metal may be moved paralled to the axis of the 
section so long as the section is not distorted to such an extent 
that it will break by twisting. Any change of metal, however, 
toward or from the axis of the section, changes the modulus and 
hence the resisting power of the section. 

Fillets may be added at the interior corners giving a shape 
similar to most frame tops. 

For the bottom, a slight deflection or slope of the web, as shown 
by the dotted lines, gives a result very similar to a plain cast iron 



258 



MACHINE DESIGN 



a b 


a' 


b 






j 


d c\ 

/ 
/ 

y J 


1 

/ 

/ 
/ 


'2' 




/ 





b' 



engine or lathe bed. Other minor changes such as slight curves 
instead of straight sides might be made without any loss of 
rigidity. In any case where the shape of the simple section is 
found and the designer wishes to increase the thickness of any 

part he may do so and the result is merely 
to increase the factor of safety. 

Suppose some other than a uniform 
section is desired, the same process would 
be employed in finding the stresses as 
given above. The modulus, Z, however, 
would be obtained as shown in Art. 144. 

If under very heavy loads it is advisable 
to specify one or more steel I beams or 
channels from Cambria, this may be done by 
making a trial selection of a section and substituting the value 
of Z and A in the formulas as before. If this value /S, ^r&i — ^t 
= 8000 to 16,000, the exact value depending upon the rigidity of 
the beam, the condition is fulfilled as in the case of the cast frame. 



Fig. 123. 



136. — The final determination in this design is to obtain the 
length of the frame to prevent overturning when the load is 




applied. Let W ^ Fig. 124 = the weight of the frame, then from the 
force diagram we have the following moments about the end at 6 

TFa; + TF% - (T7'^ + TT,?,) - TF Js = 
but W\l, + WJ,, = W'% and Wx = W,k. 

The length may then be obtained by adjusting the values of x 
and Z5 such that the equation will be satisfied. 



TOGGLE-JOINT DETAILS 259 

To obtain the length, however, in a more direct way the fol- 
lowing can be used : 

If x = l-l^ andZ5-Z2-^2 then W {I -l^) =W ^1,^-^2. 

Knowing the cross-section of the bed in square inches, the 
weight of 1 in. in length would be .26^; the total weight of the bed 
being .2^l^A approximately. ^ Then l.^^==W{l -l^) -^.13A. 

Let 1^ = 1^— a where a is the offset as shown, then l^ = W 
(I —l2 + a) -^.13A, from which we obtain the formula 



Z,= - 3.85^ + ^ 7.7 (l + a) -^ + 14.82 (^^j 

^The weight of the frame Wr, would be greater than here shown be- 
cause of the metal in the ends of the frame and the attached mechanisms, 
all of which would be effective. The error, whatever it may be, is toward 
that of safety. 



260 



MACHINE DESIGN 




First Alternate, Design No. 1 



'm m^ 



w 




M^-~ l^____ 



taj 



:b 



J 



Fig. 125, A 
The Toggle Joint Press 

137. Assignment. — 

W= . . . .; l=. . . .; r= . . . .; 6 (min.) - . . . . 

In this design the lever is placed within the bed rather than 
above it. It will be noticed that the end of the bed is slotted to 
allow for a movement of the lever arm between the points x and 
x\ The weakening of the bed due to this slot need not be con- 
sidered a serious matter. With a long and shallow bed, however, 
the movement of the arm will be small and will give a very slight 
movement to the sliding block. For our purpose this machine 
may be designed merely to exert a pressure between the two 
sliding blocks, in which case a very slight movement is all that is 
necessary and the form shown will be satisfactory. 




Fig. 125, B. 

In case the movement of the sliding block is desired greater 
than that allowed here, the lever may be arranged as shown in 
Fig. 125, B. 



ALTERNATE DESIGNS 



261 



Second Alternate, Design No. 1 




Fig. 126. 
Vertical Hand-power Press 

138. Assignment. — 

W = . ... ',1= .... ',V =....', 6 (min.)= . . . 

This design follows the principles laid down in No. 1, with two 
exceptions. First, the length V here becomes so small that a 
separate crank cannot be used and a bent shaft or an eccentric is 
substituted. In the eccentric the length V is the distance between 
the center of the shaft and the center of the eccentric. Second, 
the thrust of the sliding block is received through a screw 
directly against the base of the frame. A hollow rectangular 
section is suggested as the best shape of the frame. Investigate 
also for the screw and nut to resist the thrust. 



262 



MACHINE DESIGN 
Third Alternate, Design No. 1 




Fig. 127. 
The Vertical Foot-power Press 

139. Assignment. — 

W = (100 or less) lb. 

I = (60 to 72) in. 

l'^ (3 to 6) in. 

6 (min.) = degrees. 

This machine can be used for all kinds of light press work 
where but a small movement of the ram is needed. Where this 
movement is desired as great as possible, increase V and decrease 
I, also reduce the length of the toggle members. 

The ram may be made rectangular in section and the forming 
dies need not be developed. The frame is hollow and the lever 
I is fastened on the plane of the toggle. 



ALTERNATE DESIGNS 



263 



Fourth Alternate, Design No. 1 




X 



Fig. 128. 
Small Hand-power Punch 

Fig. 128 shows a small bench tool, used for punching sheet 
iron and other thin metals. Because of its simplicity only two 
parts of the assignment will be given. All other necessary 
assumptions may be made by the designer and a complete set 
of calculations and drawings made. The diagram to the right 
shows the mechanism. 

140. Assignment. — 



W = (at end of lever I, 50 to 100) lb. 

T = (length of throat) in. 



264 



MACHINE DESIGN 
Fifth Alternate, Design No. 1 




Fig. 129. 



Hand-power Punch and Shear 

The hand-power punch and shear is strictly a bench tool for 
operating on light work. The force at the end of the lever arm I 
should not be greater than 100 lb.; V is the eccentricity of the 
cam, a is the distance from the pivot point of the shear arm to 
the point where the cam force is applied, and 6 is the distance 
from the pivot point to the point of greatest shearing resistance. 

141. Assignment. — (See Design No. 2 for methods.) 

Kind of material to be cut 

Length of cut or diameter of punch in. 

Thickness of plate to be cut (up to |) in. 

Depth of throat in. 



CHAPTER XIV 
DESIGN OF BELT-DRIVEN PUNCH OR SHEAR 

142. General Statement. — A belt driven punch or shear is the 
machine selected to represent the second general design. In- 
cluded within this one machine are problems covering the design 
of frame, levers, gears, fly-wheel, pulleys, bearings, shafts, 
sliding head, punch, die, clutch, stripper and cam. The fact 
that this machine finds such general use in manufacturing plants 
and that it embodies such a variety of designs makes it an ideal 
subject for analysis. Fig. 130 shows a motor driven shear of 



.»'- iiH^ i.'_ • 




Fig. 130. 



late design. It is not expected that the required design will be 
for a motor drive, but that the distance between the bearings be 
shortened and pulleys used instead. In giving out the design 
the following requirements will be made: first, the work to be 
accomplished, i.e., diameter and depth of hole to be punched or 
the cross-section of the piece to be sheared; second, the maximum 
distance from the edge of the plate to the center of the cutter, 
i.e., the depth of the throat of the machine; third, the average 
cutting velocity of the punch or knife in inches per second, 'or 
the r.p.m. of the cam shaft. 

265 



266 



MACHINE DESIGN 



In the analysis of the methods employed in working up such a 
design, the frame sections will be carried out somewhat in detail 
because of the advanced character of the work; the rest of the 
machine will be dealt with more briefly. In making the assign- 
ments, the members of the class should be given values that 
differ materially from those worked out here. The five sample 
plates at the end of the design show a complete set of drawings 
of such a machine. 

143. Requirements of the Design. — A machine to punch a 1-in. 
hole through f-in. mild steel plate, the center of the hole to be 
not greater than 7 in. from the edge of the plate. The velocity 
of the punch during cutting may be taken in this case as approxi- 
mately 1 in. per second. 

144. Frame. — The material used in the frame of such a 
machine is either close grained cast iron or steel casting. The 
general shape is about as shown in Fig. 130 and the sections of the 
frame. Fig. 131, are either hollow cast iron as shown in B and C 
or web-shaped steel as shown in A. Of the three sections, B and 
C are the most common. Fig. 137 represents the outline of the 




Fig. 131. 



assembly drawing as finally worked out about x x, the center line 
of the frame. To plan the general shape of the frame about the 
punch, begin by laying off the throat depth, G, say 8 in., along 
the line x x. Find H of the same figure by assuming some shape 
of frame section and calculating the sizes for the various parts 
of the section as described in this article. Find also other safe 
sections at various angles to the horizontal and trace the outer 
curve of the frame through the outer points of these sections, 



PUNCH AND SHEAR DETAILS 



267 



after which plan the speed mechanism and locate the shafts. 
It is necessary many times to modify the first layout a great 
deal but this must be expected and should not cause 
discouragement. 

To work out the sizes of the horizontal section along x x, select 
the shape, say B, Fig. 131, from the standard forms and apply 
the method used in Art. 135, taking G as the depth of the throat 
and y as the distance from the edge of the casting to the center 
of gravity of the section. 

In applying the formula /Sj = —^ exercise care in obtain- 

^{t or c) 

ing a satisfactory value for Z in the unsysmmetrical section. To 
get Z it will be necessary to determine the moment of inertia I of 
the section and then find Z by the following : 

for tension Zt= - for compression Zc = —r 

y y 

Make a trial selection of some sizes for the section and find the 
gravity axis by cutting out a pasteboard section and balancing 
it upon knife edges; or a better way is by the following: assume 




any line of reference as ah, take the algebraic sum of the moments of 
each rectangular section about this line of reference and divide by 
the total area; this will give the distance x between the line of reference 
and the gravity axis gg of the section, ab may be taken at any 
position in the section but the work will be much simplified if it 
is taken at the edge or at the center of the section. When gg 
is determined find / by the following : to the sum of the products 



268 MACHINE DESIGN 

of each area hy the square of the distance from its center of gravity 
line to the gravity axis of the section add the moment of inertia of 
each section about its own gravity axis. It will be remembered that 
the moment of inertia of any rectangle about its own gravity axis 
is / = hh^ -T- 12 where h = the total height of the section. 
Assume the section with sizes as shown in Fig. 132 then 

_ 70X11.5-2X10X10 _, Q. 
^~ 70 + 20 + 54 -^-^in. 

/=70X(7.3)2 + 54X(4.2)2 + 20X (14.2)2 + 
14X(5)3 , 3X(18)3 , 10X(2)3 



12 ' 12 ' 12 



= 10326 



^ 10326 ,^_, . ^ . 
Zt= f. Q = 1054 for tension 
y.o 

„ 10326 „_, 

^c= -I r 9 =679 for compression. 

W is the pressure on the punch in pounds. If the ultimate 
shearing stress of mild steel be taken at 55,000 lb. per square 
inch, W would be 129,591 lb. Considering the trial section only 
on the tension side, since this is usually the weak side of the 
section, we have >Si + 1^3 = aS^ = 900 + 2189 =3089 lb. per square 
inch. This fiber stress would be large for cast iron in tension, 
hence another section must be selected. 

Take for a second trial the section Fig. 133, we have, if worked 
as above 

0^ = 2.97 in. 

7 = 21,049.44 

7— ! 1680 for tension 

\ 1317 for compression. 

Si+S2 = St = 2154: lb. per square inch. 

In like manner we should work out the compression side by S^ — 
S2 = Sc. The algebraic sum of the two gives 571-2013 = - 1442 
lb. per square inch. The sign may be considered either positive 
or negative since it merely indicates the direction in which the 
force acts and does not affect the magnitude of the force. See 
also Art. 135. 



PUNCH AND SHEAR DETAILS 



269 



Any other section of the frame can be determined by finding 
S2 in the manner shown above and combining with it the value of 
Si = W cos a -i- area. The value of S^^ is a maximum when a is 
zero and becomes zero when a is 90°. It will be seen, Fig. 134, 




"—A 



Fig. 134. 



that at the section A, aS^ = T7 -j- area A; at 5, Sj^ = Fb^Sivea B, 
but Fb = W cos a hence S^ = W cos a-^area B; at C, S^ = W cos 
a -^ area C and so on until S^ becomes zero at section E. At this 
point the frame should be examined for both bending and shear- 
ing and the larger requirement taken. In all probability section 



270 



MACHINE DESIGN 



E will be made larger than the calculated size to accommodate 
the finishing around the head. It will be satisfactory in this 
design if we obtain sections at a = 0, 45 and 90 degrees. 

To find any section, say a = 4:5 degrees, determine the height of 
the gap, k, and draw the outline of the gap. The value k is con- 
trolled by the space taken up by the dies, metal to be punched, 
and clearance. It cannot be determined exactly, but a good 
estimate may be made. Assuming some section of the frame 
as Fig. 135 and solving for the fiber stress as before, we find 
X =2.67 in. 
1 -15,655. 
Zt = 1382. 

8^+82 = St = 1872 lb. per square in. 

Note. — In finding MmM = SZ the lever arm varies, depending 
upon the cosine of the angle a. 

Investigate also for Sc. 




Fig. 135. 



For the vertical section take Fig. 136, in which case we have 
a; = 3.26in. 
7 = 4411.79 
Zf = 655. 

S2 = 791 lb. per square inch, which shows that the section 
could be materially reduced in size if it were desired. The 
reduction could very properly be made according to the dotted 
lines. If it were considered necessary, this section should also be 
investigated for compression. 

To investigate for shearing on the vertical section we have, 



PUNCH AND SHEAR DETAILS 



271 



allowing the shear to be absorbed by the entire section of 136 
sq. in., 

^ 129591 „.„,, . , 

/Ss = — T^^ — = 953 lb, per square inch. 




Fig. 136. 

145. Having determined several important sections in the 
frame, the outline of the G part of the frame can be drawn. This 
outline will of course be modified somewhat for the shaft, head 
and leg. 

It will be noticed that a somewhat higher fiber stress has been 
allowed in the material for this frame than in the material used in 
the frame of the first design. This is about as would be expected. 
Any casting planned to fill a very important place in the design 
of any machine would be made of the best close grained gray 
iron. It is advisable to keep the size of this frame as small as 
possible consistent with strength and, since the best of cast iron 
would have an ultimate strength of 25,000 to 30,000 lb. per 
square inch, it would be considered safe to allow a fiber stress of 
2000 to 2500 lb. per square inch, which corresponds to a factor of 
safety of 12. 

The shape of the section may be varied to suit the conditions, 
from a large and thin section as here treated, to a small compact 
and possibly solid section. The latter condition prevails in some 
machines where the gap is long and the main section would be 
necessarily crowded into the smallest space. 

Steel cast frames are very common, especially on the larger 
machines. When made of steel the frame section may be made 
much smaller. ^f = 12,000 to 15,000 lb. per square inch. 



272 



MACHINE DESIGN 



Tension bars are provided for machines with long gaps, 
bars are very necessary when doing heavy duty. 



These 




^—D 



Fig. 137. 



146. The Maximum Punching or Shearing Force is used in 
calculating the frame sections. The ultimate shearing stress of 
the metal multiplied by the area to be cut gives the maximum 
load on the punch or flat shear. If the maximum load on a 
hevel shear is desired, multiply the maximum load on a flat 
shear by the following: 



THICKNESS OF THE METAL 





1 


1 


1 


H 


H 


If 


H 


If 


If 


1| 


4° Bevel 


.42 


.48 


.54 


.61 


.67 


.73 


.79 


.85 


.92 


.98 


8° Bevel 


.23 


.3 


.37 


.44 


.51 


.58 


.65 


.73 


.81 


.88 



,95 



Look up articles on the Shearing of Metals in the American 
Engineer and Railway Journal, Vol. LXVII, page 142. 

In any machine of this kind it is safe to allow 15 to 20 per cent 
for the friction of the parts while performing the heaviest duty. 



PUNCH AND SHEAR DETAILS 



273 



The total pressure to be accounted for at the driving end in this 
machine will then be 129,591-^.85 = 152,460 lb. If the eccen- 
tricity of the cam be taken the same as the thickness of the 
thickest metal to be punched, =} in., the twisting moment on 
the main shaft will be approximately |X 152,460 = 114,345 
inch pounds. 

147. Working Depth of the Cut. — The actual cutting depth 
(depth of penetration) of a punch or flat shear may be used in 
determining the foot pounds of work done at the tool, and is a 
certain percentage of the total thickness of the metal. Generally the 
tool in its movement passes entirely through the metal, but the 
work of cutting is finished when the tool arrives at the depth of 
penetration. This percentage varies somewhat with the kind 
of the metal, but for mild steel it has been found by experiment 
{Am. Mach., Oct. 12, 1905) to be 



_3_ 

16 



3 
32 



Thickness of metal, in inches 1 I f i f t^( 

Depth of penetration in per cent of 

plate thickness 25 31 34 37 44 47 50 56 62 67 75 87 

Thus the work of cutting is finished when the punch (or flat 
shear) has reached a depth of .25x1 = .25 in. in a 1-in. plate, 
.185 in. in a ^-in. plate, .125 in. in a J-in. plate, and so on. 



148. Diameter, Width and R.P.M. of the Pulleys.— Table 
LVIII gives values agreeing fairly well with current practice for 



TABLE LVIII 



Machine will punch 


Diameter of pulley 


R.p.m. 


iin.Xiin. 


10 


200 to 250 


J in. X ^ in. 


12 


200 to 250 


1 in.Xf in. 


16 


175 to 200 


1 in. X 1 in. 


18 


150 to 175 


2 in. X 1 in. 


30 


150 to 175 



274 MACHINE DESIGN 

the diameter and revolutions per minute of the pulleys. To 
determine the width of the pulley face, or the width of the belt, 
no definite rule can be stated. Practice varies between a 2-in. 
belt on a ^-in. X i-in. machine, and a 6-in. or 7- in. belt on a 2-in. 
X 1-in. machine. Calculations for belt sizes on such machines do 
not give very satisfactory results because of the small percentage 
of each revolution that the machine is actually working. It is a 
good experience, however, if each man would apply a few trial 
conditions and note the results. First find the effective pull P 
on the belt, by the horse-power formula or by moments from the 
cam shaft, assuming the punch or shear to be cutting full value 
all the time, and then take the percentage of this which is repre- 
sented by the proportion of the total time that the cutter is 
actually working. Figure the belt from this result as in Art. 
117. In all probability, catalog sizes will finally be taken. 

149. Fly-wheel. Weight. — The weight of the fly-wheel may 
be obtained by either one of two methods; first, by assuming 
the wheel, when running at full speed, to have stored up energy 
enough to do a certain definite work; second, that the wheel shall 
have only a certain allowable fluctuation from full load to no load. 
From the first method, a fly-wheel for a machine of this kind may 
be designed to fulfill a number of conditions, from a wheel such 
that its kinetic energy will just equal the energy absorbed by the 
machine during punching (in which case if we disregard the belt's 
action, the velocity of the wheel would become zero after each 
hole punched) , to a wheel of such a size that the residual energy 
will be sufficient to keep the speed fairly constant. Current 
practice approaches the former and in this consideration will be 
adopted. 

Having given the force to be accounted for at the driving end 
as 152,460 lb., assume that this force acts through, say a maxi- 
mum of one-half the total depth of the plate, f in. or ^\ ^^-j then 
the energy exerted would be 4764 foot pounds. Apply the 
formula Wv^^2g to the mean rim diameter, where 17 = weight 
of wheel in pounds assumed centered at the center of the rim 
and 2;= velocity at any point in this circumference in feet per 
second. Assuming 36 in. as this diameter with 150 r.p.m. (see 
Art. 150) we have 



PUNCH AND SHEAR DETAILS 



275 



Wv' 



= 4764; TF = 553 1b. 



2g 

The depth of penetration in this application is not used according 
to the table. This should not be confusing since it is merely for 
illustration. 

Find the weight of the fly-wheel also from some acceptable 
formula based upon the fluctuation of speed, using for the allow- 
able fluctuation 20 to 25 per cent, and check with the above. 

Arm. — The fly-wheel arm may be calculated as follows: 
estimate the time required in punching one hole, then find the 
distance through which a point on the center line of the rim will 
move during this time; this will be the value V in PF = 4764. 
Since the shaft is running 15 r.p.m., each revolution will take 




Fig. 138. 



four seconds. Assuming the velocity of the punch during 
action to be the same as that of the cam center we have 3.1416 
X 1.5 -T-4 = 1.1781 in. per second. The time occupied in punching 
is I ^ 1.1781 = .318 second. The velocity of the rim of the wheel 
is 1413.7 ft. per minute = 23.56 ft. per second, from which we 
find that the rim will travel 7.5 ft. before stopping. 
Applying PV = ^764c, 
P = 635 1b. 
The value of P may be found in another way. First, with the 
radius of the large gear = 22.5 in., find the force p between the 
gears, Fig. 138. From the moments around the cam shaft, this 
is 



276 MACHINE DESIGN 

129591 X3 _,082 1b 

and by moments around the driving shaft 

p 5082X9 _^.,, 
P= —. — t7^=63d lb. 
4X18 

Having found P, the tractive force due to the stored up 

energy of the wheel rim, obtain the large dimension of the arm 

Pr 
at the center of the shaft by the formula -^rT = .O^h^S, If A^, 

the number of arms, = 6 and S = 1500, 



'4 



635X18 _o' 
— o m. 



6X. 05X1500 



A low fiber stress is used because of unknown stresses that are 
apt to be in the casting. Straight arms are preferred to curved 
arms and they should have well-rounded fillets next to the hub 
and rim. The section of the arm near the hub and that at the 
rim are always similar. The dimensions at the rim should be 
taken not less than two-thirds of the corresponding dimensions 
at the hub. The ordinary arm has the thickness at the center 
of the section about one-half of the length of the section. The 
radius of the side of the section is about three-fourths of the 
longest dimension of the section. The value b as given in the 
formula is sometimes taken at the center of the shaft and fre- 
quently at the edge of the hub. This, it will be seen, makes 
very little difference in the average pulley. 

150. Driving Shaft. — If the bearings are close to the pulley 
and gear the bending will not be excessive and the shaft may be 
figured with a low fiber stress merely to resist twisting. Taking 
aS = 6000, the diameter of the shaft will be 2.2, say 2{ in. 

On machines where the pull of the belt and the side thrust 
from the gears are fairly great, also when the bearings are far 
apart, it is necessary to design the shaft for combined twisting and 
bending. In such a case find the side thrust due to each, the belt 
and the gears, and calculate the shaft from the bending moment 
as a beam fixed at the ends and loaded at two points. See 
Art. 4. 



PUNCH AND SHEAR DETAILS 



277 



In locating the shaft DD, it is first necessary to have the 
approximate position of the main shaft and the diameters of the 
gears. Knowing the angular velocities of the two shafts the 
diameter of the small gear may be assumed and the distance 
between the shaft centers obtained. In this machine if the 
cutting speed of the punch is 1 in. per second, the center of the 
cam will travel approximately 60-^4.71 = 13 revolutions per 
minute. Calling this 15 and the revolutions per minute of the 
pulley shaft 150 the ratio of the gears is 10. With 4^ in. as the 
diameter of the pinion, the shafts will be 24| in. between centers. 

151. Gears. — Design according to Art. 92 for machine cut 
teeth. The pinion should be shrouded. The diameter of 4J 
in., as used in Art. 150, is merely for illustration. This value 
would be rather small for the construction of a perfect tooth. 
The arms of the large gear are similar to those in the fly-wheel 
excepting that the driving force will be absorbed by not more 
than one-half the number of arms in the gear. 

152. Main Shaft. — The main shaft or ''cam shaft" as it is some- 
times called would be made of hammered steel. Figs. 139 and 
140 show two common forms. B^, B^ and B^ are journals, and 
C is the cam which operates the punch. The greater part of the 
thrust from the punch is absorbed at the journal B2, B^ being 
added for the double purpose of reducing the strain of the shaft 
and for an outside connection for adjustments. 



-X- 



B( 



^d- 



C 



'6* 



Bi 




Fig. 139. 



In designing the shaft the part a may be figured to resist the 
twisting moment due to the thrust on the gear x, allowing a fiber 
stress of, say, 6000 lb. per square inch for shear. It will be 
noticed, however, that the thrust on the gear produces a bending 



X 



moment on the shaft, the lever arm being ^- This bending 



278 



MACHINE DESIGN 



moment may be of such magnitude as to make it necessary to 
use the combined formula. It would be well to obtain the 
diameter from both formulas and check them. 

The length of the journal may be taken from 2 diameters to 
2.5 diameters of the shaft. The length of b will be quite variable 
and will be governed by the frame of the machine. The diam- 
eter of b will depend upon the judgment of the designer. In 
some shafts it is made equal to the diameter of the left journal 
while in others it is enlarged to the size of the main journal. A 
high speed machine would require a larger and stiffer shaft than 
a slow speed machine, because of the heavy shocks to which the 
shaft is subjected, hence the diameter of b would be as large as 
possible. 

Take the size of the main journal such that the pressure per 
square inch of projected area will not exceed 3000 lb. assuming 
the entire thrust from the punch to be taken up by it and that 
of the cam not to exceed 8000 lb. Lower values than these are 
desirable, especially on the cam, where 5000 lb. per square 
inch of projected area is a good value. It will be seen from the 







)i 


c 






Yf 


B, 












Fig. 140. 



above that the projected area being constant, a bearing may be 
changed in shape decidedly and yet give good service. As an 
illustration B^ may be long and slender as in Fig. 139, or short and 
thick as in Fig. 140, so long as the shaft at this point is stiff 
enough to resist bending and shear. Conditions within the 
machine itself usually determine the shape of bearing and cam. 
When the sizes are approximately determined, they should be 
constructed graphically to scale, usually having the two surfaces 
continuous along one line as at x. 

The cam varies from 3 to 6 in. in length, and from 6 to 12 in. 
in diameter. The diameter of the bearing in such a case is 
governed somewhat by the eccentricity of the cam. 

The cross-sectional area of the bearing B^ along its outer face 



PUNCH AND SHEAR DETAILS 279 

next the cam must be sufficient to resist the effect of shear; it 
must also resist the bending moment produced by the thrust 

multiplied by the half length of the cam I - j and the torque 

produced by the thrust multiplied by the eccentricity of the 
cam. This should be worked by the combined formula, remem- 
bering that B^, where used, would reduce this bending moment 
somewhat. 

In machines where the distance between B^ and Bg is great 
there is a bending of the shaft between the bearings. This is 
especially true where B^ is omitted as in some horizontal machines. 
Such a condition is equivalent to a beam in flexure with the 
reactions at B^ and C and the applied load at B^. The effect, 
however, is not the same in the calculations as a simple beam 
because of the support given to it by the boxes. 

It is safe to assume that the bearings are sufficiently loose to 
allow some bending, but not loose enough to consider it as a 
simple beam. Probably a safe assumption would be 50 per cent 
of the maximum load applied at the cam center and resisted at 
the bearing centers as supports. 

The frame should be fitted with a phosphor-bronze bushing J 
in. to I in. in thickness surrounding the journal B^. This 
bushing is made a forced fit with the frame. 

The sizes of B^ would vary between 2 in. and 4 in. for both 
diameter and length. 

Application. — Calculating the shaft for twist at its smallest 
diameter, at the gear, gives (i = 4.59, say 4.5 in. 

The cam diameter, assuming a length of 4 in. and a pressure 

per square inch of 5000 lb. is ,^„„ — 7 = 6.5 in. 
^ ^ 5000x4 

B2 will then be 5 in. diameter and, if we allow 2500 lb. per 

129591 
square inch projected area, will have a length of — ^ = 10.4 

in., say 11 in. 

B^ may be taken 2^ in. long by 3 in. diameter. 

153. Sliding Head. — Of the different types in use, two of the 
very common ones are shown in Figs. 141 and 142, the former 
being used in the smaller machines. The chief objection to the 



280 



MACHINE DESIGN 



bronze block is its liability to wear unevenly thus causing lost 
motion and an irregular movement of the block while punching. 
In the latter form, the entire thrust is carried on a hardened steel 
block set into the cast iron sliding head and the wear, if any, is 
practically uniform. The size of the bearing surface in the steel 




Fig. 141. 

block may be obtained from the crushing strength of the steel 
casting. If this value be taken at 90,000 lb. per square inch 
with a factor of safety of 6, the projected area of this bearing 
will be 129,591 -i- 15,000 = 8.6 sq. in., from which, if the length 

of the cam be 4 in., the breadth of the 
bearing will be 2.15 in. say 2^ in. The 
breadth of the sliding head face will be 
seen to depend upon the construction 
of the vibrating arm. Make the vibrat- 
ing arm a steel casting and allow from 
j in. to 1^ in. at x, and a small clearance 
at y. This part of the work must be 
done graphically. The values a and h 
will depend respectively, upon the width 
of the frame and the diameter of the 




Fig. 142. 



bolts used. 



154. Clutches and Transmission Device. — In operating any 
machine having an intermittent motion a clutch is commonly 
used to serve as a connector between the power supply and the 
work. The application of the clutch to the simple punching or 
shearing machine is shown in Fig. 143. It is usually applied 
directly to the hub of the large gear and is operated through a 
system of levers and cranks by either hand or foot. When the 
punch is not operating, the large gear, which is designed with a 
long hub to act as a bearing, runs loose, the shaft remaining 



PUNCH AND SHEAR DETAILS 



281 



stationary. The clutch sleeve slides on the shaft over a splined 
key and when the punch is to be operated this sleeve is thrown 
to engage with the corresponding part on the gear hub. When 
the hole is punched a counterweight brings the sleeve back to 
its former position and the movement of the punch ceases. 

Clutches are formed each having two, three, or four j aws. These 
jaws may be formed as a part of the wheel hub as shown at A 
and B, cast from steel and bolted to the fiat face of the wheel 
hub as shown at C, or cast from steel and fitted to the interior of 
the wheel hub as shown at G. In heavy work C and G are 
preferable. 

That part of the clutch subjected to the greatest wear is the 
front face of the jaw. This is sometimes fitted with a plate of 
high carbon steel which can be replaced when necessary with a 
new one. The rear face of the jaw is usually perpendicular to 
the front face of the wheel but is sometimes cut to an angle of 
30 to 45 degrees. There should be sufficient clearance between 
the jaws on the sleeve and the wheel to enable them to be easily 
thrown together while in motion. This should be from | in. 
to i in. 

The clutch sleeve may be shaped as shown in either D or F. 
The following sizes, table LIX will meet average requirements. 







TABLE 


LIX 






Shaft = 


2 in. 


3 in. 


4 in. 


5 in. 


6 in. 


a 


1 


1 


U 


n 


If 


b 


1 


1 


U 


n 


If 


c 


= 


/ 


— 


(a + b) 


d 


4 


5i 


7 


9 


m 


e 


5 


7 


9 


11 


12 


f 


3 


4 


5i 


7 


8 


g 


1 


1 


u 


If 


n 



282 



MACHINE DESIGN 



There are two general methods of designing the transmission 
device; the first and simpler one E having the clutch between the 
gear and the frame, and the second H, having the gear between 




CLVTCH. 



PUNCH AND SHEAR DETAILS 



283 



the clutch and the frame. The latter method necessitates a 
hollow shaft in order to obtain a rigid connection between the 
sleeve and the clutch and is not much used on small machines. 



155. Punch, Die, and Holders. — In all punch and die work the 
die is made a little larger than the punch for clearance. The 




A 



SLI^i NO H e.AZ>. 



V. 



I ! 



T 



■i«f 



¥ 



B 



SLirHUG HEAH. 



r^i 



¥ 



r> 



s 




a 





Fig. 144. 

action of the punch on the material is shown in A, Fig. 144, the 
hole tapering from the size of the punch on one side to the size 
of the die on the other. This taper is slight and is considered of 
no consequence in rough work, but in finished work it is a difficulty 
that can easily be remedied by reaming the hole afterward. For 
reference see " Dies, Their Construction and Use. " Woodworth 



284 MACHINE DESIGN 

There are various methods of fastening the punch to the 
sliding head; B shows the bottom of the sliding head fitted with 
the square ended socket and punch. A screw ended socket is 
sometimes used as at E. C shows the bottom of the head flanged 
and drilled for the attachment of either punches or shears. In 
single machines it is desirable that both punching and shearing 
be done. Where such is the case this is a good form. Side 
adjustment of the punch may easily be made if the head be 
slotted as at D and fitted with a tee block as E. Dies are made 
from high carbon steel and are held in a holder; the holder in 
turn is bolted to the horizontal face of the frame. A certain 
amount of adjustment is necessary in locating the die, conse- 
quently the holder is made in two parts. 

Other Types of Shearing and Punching Machines 

The smallest sizes of punching and shearing machines are oper- 
ated by hand power or foot power, medium sized machines are 
operated almost exclusively by belt and the largest machines are 
operated by belt, steam, water or electricity as shown in Figs. 
145, 146, 147, and 148 respectively. These designs show 
present practice and are added to enable the designer to become 
more familiar with the form of the parts and the make-up of the 
machines in general. 

It will be noticed that in the larger machines the frame is of 
such a size as to project below the floor, the weight being carried 
on legs or lugs cast on the side of the frame. It will also be 
noticed that arrangements are made at the top of the frame for 
the attachment of a crane to assist in handling the material. 

Most single machines have the lower end of the ram so con- 
structed that either punches or shear blades may be attached. 
This requires some little time in changing and adjusting the tools. 
Double machines avoid the necessity of such changes. 

Machines such as are here represented require more work than 
should be expected of one assignment. They may, however, be 
assigned to two men. This is especially true of the double 
machines, in which case the frames may be worked up independ- 
ently, and the driving mechanism, jointly. Electric motor sizes 
and capacities may be obtained from any standard catalog of 
electric machinery. 



LARGE PUNCHING AND SHEARING MACHINES 285 




Fig. 145. 




Fig. 146. 



286 



MACHINE DESIGN 




Fig. 147. 




Fig. 148. 



TYPICAL DRAWINGS 



287 



fCL 



^^^ 




m M m 



"CD CD OJ 




X 

o 

z 

Q. 

CE 
UJ 

$ 
O 
0. 






UJ < *5^ 

-J § 

03 <i. 



i 



«>8 







288 



MACHINE DESIGN 




.]- 



-?*- 



O 
Eh 











^ 



-^H;;?'^ ■''* tj»tJ^o/'Jfu 



L 



\ \v 

1 1 
1 1, 




TYPICAL DRAWINGS 



289 




290 



MACHINE DESIGN 



00 

O 

PL, 




PLATE C 7 



■J—V-^ ^-M 



!-• 



- / -7^* 



h'V^'-^..' 






u 



;* 



^1 



fomf£ Jeep fcr 

J'/J'ca^ aerew. 

t tfe^vi'reel 



Ort'/I S -tmpfor 
iH'SifHj Server 

Top 1^ 'det^for 

ixzj e<t/>ser€>r 

4 tfe^vtrrd 







Core \ for i^'founa 




SINGLE POWER PUNCH 
DETAILS 

Pvrtfue University Lafayttte IneL 
uriAfXf. . -^ - 



. 



ALTERNATE DESIGNS 
First Alternate, Design No. 2 



291 



^^u\^«^» ^\ 




B 



m . m\ 

I 

c 

Fig. 149 

The Bevel Shear 
(Niles-Bement-Pond Catalog) 

156. Assignment. — 

Kind of material to be sheared 

Width of plate to be sheared (6 to 12) in. 

Thickness of plate to be sheared (J to 1) in. 

Depth of throat (6 to 18) in. 

Strokes of the ram per minute (15 to 20) 

The frame sections may be calculated, if desired, to a regular 
outline as shown in the dotted lines, after which modifications in 
this outline may be made by approximation. A better way, 
however, would be to sketch the approximate longitudinal frame 
section as above and figure for each of the several irregular 
sections, as A, B and C, 



292 



MACHINE DESIGN 



Second Alternate, Design No. 2 




Fig. 150. 



Horizontal Power Punch 



(Niles Tool Works Co. 1900 Catalog) 



157. Assignment. — 

Kind of material to be punched 

Size of largest hole punched in. 

Thickness of the plate in. 

Distance of center of hole from edge of plate in. 

Number of holes punched per minute 



Horizontal punching machines may be designed in the same 
general way as the one described in the notes. It will be found 
that the frame sections may be calculated in the same way 
although the frame not being so regular will require a little more 
care in selecting the shapes and sizes of the various parts of the 
sections. 

Machines of this type usually have a more shallow throat than 
the vertical type. 

The line of the punch center may be raised from the center of 
the ram to the upper edge and is found convenient when punching 
near a shoulder. 



ALTERNATE DESIGNS 
Third Alternate, Design No. 2 



293 




Fig. 151. 
The Bulldozer 

158. Assignment. — 

Length of stroke (6 to 18) in. 

Maximum pressure (5000 to 30,000) lb. 

Number of strokes per minute (10 to 15) 

The Bulldozer, one of the most powerful of the horizontal 
presses, is used in forming or squeezing metals to shape between 
large dies in such processes as upsetting and bolt heading. It is 
also occasionally used in punching and straightening. The dies 
are very heavy, and sometimes the stroke is made long enough 
to permit a number of dies being inserted at one time, so as to 
allow several operations on the specimen without reheating. 

Assume a typical work card, having the ordinates represent 
total pressures in pounds and the abscissas represent per cents of 
stroke. Let the work to be performed be such that the dies first 
strike the specimen at 25 per cent of the stroke, also let it require 
the following total pressures to complete the work: 

Per cent of stroke 25 30 35 40 50 60 70 80 90 95 100 

Per cent of max. pressure 00 60 70 75 80 82 78 75 75 80 100 



294 



MACHINE DESIGN 
Fourth Alternate, Design No. 2 




Fig. 152. 



Atlas Power Press 
(Atlas Machine Co. Catalog) 

The press shown in Fig. 152 is designed to take the place of the 
ordinary foot-press in doing light blanking, perforating, riveting, 
forming and closing. The clutch is of the standard Johnson type. 
A ball-and-socket joint between the shaft and the gate gives the 
latter a vertical adjustment of about 1^ in. The machine is 
furnished with a combination pulley and balance wheel. The 
mechanism of the machine is shown to the right. The following 
approximate sizes may be used for checking: 

From bed to gate in lowest position 6 to 7 in. 

Stroke li in. 

Distance between uprights 3f to 6 in. 

Bed surface 7 X 10 to 8 X 12 in. 

Weight of wheel 50 to 100 lb. 

159. Assignment. — 

P (crank 5 degrees from vertical, 1000 to 2000) lb. 

T (depth of gap) (4 to 6) in. 

Revolutions per minute (200 to 300) 



ALTERNATE DESIGNS 
Fifth Alternate, Design No. 2 



295 




Fig. 153. — The Lennox Rotary Shear. 

(Joseph T. Ryerson Catalog) 

(Bethlehem Foundry and Machine Co. Catalog) 

160. Assignment. — 

Kind of material to be sheared 

Depth of throat (6 to 36) in. 

Thickness of plate (i to |) in. 

Diameter of cutters (6 to 12) in. 

Rim velocity of cutters (600 to 1000 ft. per hour) 

Notation 

Shaft L is adjustable at lower end by screw H around jR as a 
pivot. See Fig. 155. 

Shaft N is adjustable in line parallel to center of shaft by nut 
T. J and are gears of same diameter. The main pulley of the 
machine runs from 150 to 250 revolutions per minute. The 
cutters at D may be set apart a distance as great as one-fourth 
the thickness of the plate; the exact amount can best be deter- 
mined by the experience of the operator. The exact force W at 
the cutters tending to rupture the frame is rather an indeterminate 
quantity but a safe value may be found by the following formula : 



W = Af=tf\ tr 



4 



1- .6834 + 



40 r 



where t = thickness of the metal to be cut ; r = radius of the cutters ; 
/=the ultimate shearing strength of the metal; and A = area of 
the metal being cut at any time, assuming the cutters to be in 
contact at the center line. 



296 



MACHINE DESIGN 




Fig. 154. 
Lennox Rotary Bevel Shear 




Fig. 155. 
Section of Shear 



ALTERNATE DESIGNS 
Sixth Alternate, Design No. 2 



297 




Fig. 156. — Sheet Metal Flanger and Disc Cutter. 
(Niagara Machine and Tool Works Catalog) 





The section shows the machine with flanging rolls. 
These may be changed to cutter rolls as shown at the 
left. Other small rollers hold the metal to the plate 
while being operated upon. Sizes of flanges obtained 
in soft sheet steel as follows: 10 to 16 guage, | to 1 
in, ; 16 to 20 gage, | to | in. ; 22 to 24 gage, J in. 
Machine will cut up to No. 8 gage and flange to No. 
10 guage. 

161. Assignment. — 

P = (1000 to 2000) lb. 

T (depth of throat on machine, 12 to 20).. . . in. 
G (depth of throat on circle arm, 30 to 40) . . in. 
Speed of the tool (15 to 20) . . .rev. per minute. 



298 



MACHINE DESIGN 
Seventh Alternate, Design No. 2 




BARTLETT & CO., N.Y. 



Fig. 158. — Boiler Head Flanging Machine Details. 
(Niles-Bement-Pond Catalog) 

162. Assignment. — This problem consists essentially of the 
development of the mechanism and the design of the parts shown 
in the detailed figures, i.e., the flanging mechanism. For applica- 
tion of these parts to the machine, see catalog. Develop the 
mechanism so that the rollers will revolve about each other with a 
uniform clearance in all positions. Assume a maximum thrust 
at the roller of (10,000 to 100,000) pounds, and design the parts 
so they will be sufficiently rigid to protect from flexure and 
breaking, and so the pull on the hand wheel will be within the 
capacity of one man, say 150 lb. 



CHAPTER XV 
DESIGN NO. 3 



Ckcck Vol' 




, /\ir Vent 



>&utomatie ReVea^e 



■jus+able Stop 




Fig. 159. 



The Air Hoist. (Whiting Foundry Equipment Co.) 

163. Assignment. — Capacity in free load lb. 

Weight of parts and friction per cent 

Air pressure (80 to 100) lb. per sq. in. 

Lift (2 to 4) ft. 

The following data may be used for checking, air at 80 lb. per square inch. 



Diam. of hoist 


Size of pipe Air consumed per 4-ft. lift 


3 in. 

7 in. 
10 in. 
16 in. 


i in. 
fin. 
1 in. 
1 in. 


1 . 17 cu. ft. 

6.63 cu. ft. 
13.50 cu. ft. 
34.49 cu. ft. 



299 



300 



MACHINE DESIGN 

First Alternate, Design No. 3 

The Allen Riveters 
(Joseph T. Ryerson and Son Catalog) 




Fig. 160. 
Lattice column type. 



Fig. 161. 
Jaw type. 



In this type of machine the piston rods connect levers of differ- 
ent lengths, thus forming a toggle joint. It very properly 
embodies features of both designs, No. 1 and No. 2. It may be 
assigned as an advanced substitute for No. 1 or as an extra. 

The following maximum pressures necessary to set rivets may 
be expected in average practice. 



Diameter of rivets in inches 
Pressure in tons of- 2000 lb. 



1 S. 3. 1 11111 11 

^ 8 4 S •'• -^^S ■'■4 ■'■2 

25 30 40 50 65 80 100 150 



ALTERNATE DESIGNS 



301 




Fig. 162. 



164. Assignment. — 

P = (8000 to 50,000) lb. 

p = (Air pressure 80 to 100) lb. per sq. in. 

r = (4 to 12) L. C. type in. 

!r = (16 to 66) J. type in. 

1= in. 

V= in. 

d (min.) = degrees. 



302 



MACHINE DESIGN 



Second Alternate, Design No. 3 




THIRD POSITION - LEVEB ACTION 
AND STROKE COMPLETED 



Fig. 163. 



Fig. 164. — Mechanism. 



The Hanna Riveter 




Fig. 165. — Development of the Mechanism. 



ALTERNATE DESIGNS 



303 




Fig. 166. 



165. Assignment. — 

p = (50,000 to 200,000) lb. 

T = (lOi to 66) in. 

Maximum movement of the die {0 + N) in. 

Air pressure lb. per sq. in. 

Assign Plunger Travel, + N 
Approximate other dimensions to table 



A 


B 


C 


D 


D' 


E 


F 


M 


N 





P 


9 


12 


1 


n 


9 


18 


26 


8f 


3i 


h 


12 



First calculate and obtain the sizes for the frame, then give lengths and 
locate levers EA, BF, CG and DH such that the first half of the piston 
movement will cause a constantly decreasing velocity of the die, and the 
last half of the piston momement will cause a uniform velocity of the die. 
As an illustration of the above: in one machine the piston movement was 
12 in., the total movement of the die was 4 in., the first 5 in. of piston 
travel gave a constantly decreasing velocity of the die through 3^ in. of the 
die movement, leaving the last 7 in. of piston movement to produce a uni- 
form velocity of the die through the last half inch of die movement. 



304 



MACHINE DESIGN 
Third Alternate, Design No. 3 




Fig. 167. 

The Alligator Riveter 
(Jos. T. Ryerson and Son. Catalog) 

166. Assignment. — (See Notation on First Alt. No. 3.) 

P = (25 to 65) tons. 

p = (air pressure, 80 to 100) . .lb. per sq. in. 

r = (9to 14) in. 

6 (min.) = degrees. 

Maximum movement of the dies (2 to 4) in. 

Assume the length of the arm of the scissors such that the force 
to be transmitted through the toggle will not be so great as to 
require too large a cylinder. Also observe that a long arm 
requires a long toggle link and hence a long piston movement. 
This style of machine is used largely in structural and car shops. 
It may be made vertical or horizontal type. The dies are adjust- 
able. The height of the gap varies from 6 to 14 in. 



ALTERNATE DESIGNS 
Fourth Alternate, Design No. 3 



305 




Fig. 168. 

MuDRiNG Riveter 

167. Assignment. — 

T7= (25,000 to 100,000) lb. 

p] = (air pressure, 80 to 100) lb. per sq. in. 

T =5 = (8 to 16) in. 

A = (5 to 8 in. 

C (min.) when 6 = degrees. 

Total die movement (3 to 5) in. 

This design may be modified by having the cylinder enclosed 
within the base if desired. In such an arrangement the piston rod 
becomes a compression member. Design also for air pipes and 
valves. 



306 



MACHINE DESIGN 



Fifth Alternate, Design No. 3 




Fig. 169. 
Lever Riveter 

168. Assignment. — 

TF = (20,000 to 60,000) lb. 

p = (air pressure, 80 to 100) lb. per sq. in. 

T = (throat, 8 to 12) in. 

A = (20 to 36) in. 

C (min.) when 6 = degrees. 

Total die movement (2 to 3) in. 

When the arms are in their inner positions the cylinder must- 
not touch them. Design also for air pipes and valves. 



ALTERNATE DESIGNS 



307 



Sixth Alternate, Design No. 3 




Fig. 170. 
25 Ton Portable. 



Fig. 171. 
50 Ton Portable. 



Hydraulic Riveting Machine 
(Niles-Bement-Pond Catalog) 



169. Assignment. — 

P = (15-50) tons. 

p = (800-1500) lbs. per sq. in. 

r=(6-15) in. 

Size of rivet (see First Alt. Des. No. 3) in. 

In this design the cylinder, frame, supports and valves are 
important in the order named. The piping is a feature that can 
be modified to suit almost any condition of frame. Such 
machines are used on structural and bridge work. 



308 



MACHINE DESIGN 



Seventh Alternate, Design No. 3 

Triple Pressure Hydraulic Riveting Machine 
(Niles-Bement-Pond Company) 

This machine is built with 
three capacities: 50, 100 
and 150 tons, for driving |-, 
1^- and l|-in. rivets. The 
gap T is made in two lengths 
12 and 17 ft. The cylinder 
is designed for three pres- 
sures of water, the highest 
being 1500 to 2000 lb. per 
square inch. By means of 
the three pressures pro- 
vided as per section (see also 
catalog) the distributing 
valve is not needed. 

On frame B is mounted 
cylinder A with main 
plunger E. To the main 
plunger can be attached, by 
means of the interrupted 
thread and nut J , the small 
plunger /. When so ar- 
ranged plungers E and 7 
move together and pressure 
on the dies is equivalent to 
the pressure p on the differ- 
ence between the two areas. 
Small plunger can also be 
attached to main plunger 
so that intermediate sleeve 
Fig. 172. 

H is locked to main plunger 
and moves with it. When 
so arranged the pressure 
on the dies is controlled by 
the difference between area 
of main-plunger and inter- 
mediate sleeve H. Cover i^ 
over die slide D contains the 
push back piston G bearing 
directly upon main plunger. 




170. Assignment. 

P = (50-150) tons. 




p = (1500-2000) lbs. persq. in. T' = (12-17) ft. 



CHAPTER XYI 

STUDIES IN THE KINEMATICS OF MACHINES. 

The following problems in kinematics are given to supplement 
the work in both mechanism and design. One or more of these 
problems may be assigned between designs 1 and 2, also between 
2 and 3 and will serve as a relaxation from the tedium of the longer 
problems of design. In their solution they contemplate pure 
mechanism (line motion) only and will not deal in any way with 
the strength or proportion of parts. The problems are arranged 
in a graduated series: first, those distinctly outlined and 
requiring little or no originality; second, those open to original 
ideas, but having one solution suggested, out of a number that 
might be made; third, those open to a number of solutions but 
requiring complete originality and invention. 

A series of illustrative problems in the study of the mechanical 
movements of machines was given in the Am. Mach., one 
problem each month, beginning December 1, 1904. In order that 
originality be developed, it is suggested that these problems be 
read in connection with the assignment given. 

The kinematic problems relating to valve gears and link 
motions are classified at the last of the list and may be given 
between designs 2 and 3 or after design 3, so as to be taken in 
parallel with or after the subject of Engines and Boilers. 



309 



310 



MACHINE DESIGN 
Kinematic Sheet No. 1 



Approximate ^ 
Epicycloid 




Appro^imcrfe 
Epicycloid 



^Exacf 
Epicycloidx 



- Exact 
Epicycloid 



Fig. 174. 



171. Assignment. — Given, a problem by which the diameters 
of the gears may be obtained. It is required to construct the 
tooth outlines for each by means of the following systems: 

Exact epicycloidal 
Exact involute 
Approximate epicycloidal 
Approximate involute. 



KINEMATIC PROBLEMS 
Kinematic Sheet No. 2 



311 




Ct . Arc o/ circle. 
b. Arc of cam. 

Fig. 175. 
Planer Cam 

172. Assignment. — 

A and C are loose pulleys, J5 is a tight pulley. 

Z> is fastened to frame of planer. 

/ moves back and forth, oscillating link F about G. 

E is rigidly connected to F by set screw. 

Levers and P are pivoted at m and n on D. 

Q and R are rollers fastened to the shifting levers. 

Pulley diameters (10-24) in. 

S^ in. 

Width of fast pulley (3-10) in. 

Width of loose pulleys, each, (2-8) in. 

Construct curve of cam so that the shifter will be constantly 
accelerated during first half of its motion and constantly retarded 
during latter half. 

During the first half of the motion of E (or F), one shifter arm 
moves outward, while the other arm remains stationary (in the 
outward position). During the second half of the motion of E, 
the second shifter arm moves inward, while the first arm remains 
stationary (in the outward position) . Place the points Q and R 
respectively directly above and below the center of rotation of 
the cam E. 



312 



MACHINE DESIGN 
Kinematic Sheet No. 3 




Cam of Home Sewing Maghine 
173. Assignment. — 

Diameter of cylinder (11-3^) in. 

Depth of groove (xe-i) ' in. 

Diameter of roller (re-f) in. 

Stroke of bar (1-3) in. 

Length of arm G (6-10) in. 

Length of arm H (10-18) in. 

Design a cylindrical cam similar to that shown in the sketch 
to engage a rocker arm. Divide the motion into 24 time periods. 
The follower is to move with a constant acceleration during four 
time periods; during the next eight periods it is to move uniformly 
with the velocity attained; during the next four periods it is to 
come to rest with a constant retardation. The return motion con- 
sists of eight time periods; during the first four periods it is to be 
constantly accelerated and during the remaining four periods it 
is to be constantly retarded. 

Required full projection of cam outline on the cylinder. This 
will require the development of the cylinder at top and bottom of 
groove. 



KINEMATIC PROBLEMS 
Kinematic Sheet No. 4 



313 




Fig. 177. 
Sewing Machine Bobbin Winder 
174. Assignment. — 

Number of threads to be laid per inch of spool length (30-100) 
Length of spool (l|-2^) in. 



314 



MACHINE DESIGN 



Kinematic Sheet No. 5 




Fig. 17 



175. Assignment. — 

Design the constant diameter cam, A, as shown, under the fol- 
lowing conditions: follower to move with harmonic motion 
from extreme right to left; to return one-half the distance by 
uniform motion; to remain at rest for one-sixth the revolution of 
the cam, and to return to starting point by uniform motion. 
Total stroke of follower in one direction = in. 



KINEMATIC PROBLEMS 



315 



Kinematic Sheet No. 6 




Fig. 179. 
Quick Return Mechanism 

176. Assignment. — 

Length of lever, A (18-24) in. 

External diam. of circular slot (8-10) in. 

Distance from center of rotating shaft, F, to center of circular 
slot (4-10) in. 

Plot velocity-time diagram of crosshead at end of arm A, which moves 
along a horizontal Une through F. 



316 



MACHINE DESIGN 



Kinematic Sheet No. 7 




Fig. 180. 



Quick Return Mechanism 



177. Assignment. — 

Radius of pin B from A (8-16) in. 

Distance from A to D (18-24) in. 

Distance of A above horizontal line through D in. 

If B revolves with uniform rotation about A, plot the velocity-time 
diagram of block at lower end of EG. 



KINEMATIC PROBLEMS 



317 



Kinematic Sheet No. 8 




i ,1 li I' 
Fig. 181. 

178. Assignment. — Lay out a Whitworth Quick Return mo- 
tion, with the path of the tool below the center B of the slotted 
crank BP, according to the following data: 

Length of stroke in. 

Length of connecting rod in. 

Length of A. P in. 

R.p.m. of crank 

Period of advance to return of tool = 2:1 

Construct the linear velocity-space diagram of the tool. 



318 



MACHINE DESIGN 



Kinematic Sheet No. 9 




Fig. 182. 



179. Assignment. — Assume center A directly above center C; 
also that slot in which B works is on the arc of a circle, with 
radius AB. Plot velocity-time diagram for member F ii AB 
rotates continuously and members are proportioned as follows: 

Length AB (6-12) in. 

Length CD (18-30) in. 

Length DE (16-20) in. 



KINEMATIC PROBLEMS 



319 



Kinematic Sheet No. 10 




Fig. 183. 



180. Assignment. — Having given an oscillating arm, pivoted 
at point B, design a cam to move the end of the arm over the 

path 1, 2, 3, 4, 5 13. The cam may have a 

uniform or varying motion while the arm may move uniformly 
or according to any law of motion desired. 



320 



MACHINE DESIGN 



Kinematic Sheet No. 11 



Uim 'i I I i ' i I I Mm I ' I — f— MB 




Fig. 184. 



181. Assignment. — Two crossheads are to be driven in paths 
AB and CD intersecting at right angles. The length of the 
stroke, CD, is one-half that of AB. Motion is to be given to 
both crossheads by a single rotating cam. Such guides and 
connecting rods as are necessary may be employed. No part 
of the mechanism is to project within the angle DAB at any time. 
Motion away from A is to be according to the following schedule: 

J stroke, uniform acceleration. 
I stroke, uniform motion. 



J stroke, uniform acceleration. 
Motion toward A to be harmonic. 



KINEMATIC PROBLEMS 
Kinematic Sheet No. 12 



321 



■ u — J 




tr ^-- 11 



" " " " '■^>>>^ i 



^P=^ 



\J 



Fig, 185. 

182. Assignment. — Let vertical crosshead be A, horizontal 
crosshead be B, the pin connection be C, then C will travel 
through the stationary cam curve as shown. 

Length of horizontal connecting rod in. 

Length of vertical connecting rod in. 

Travel of horizontal crosshead in. 

Travel of vertical crosshead .in. 

Crossheads to move out in. with uniform acceleration; 

out in. with uniform motion; out in. with 

uniform acceleration; and to move in in. with increasing 

harmonic motion; in in. with uniform motion and in 

in. with decreasing harmonic motion. 

Develop both top and bottom of groove in cone cam. 



322 



MACHINE DESIGN 



Kinematic Sheet No. 13 




Fig. 186. 



183. Assignment. — Having given the path of a groove ABG, 
a follower block is to move from A to B to C to B to A. Design 
a mechanism without the use of cams, and without allowing any 
part of the driving mechanism to extend within the angle ABC. 
Rack and pinion, or chain drives cannot be used directly to 
produce the motion. 



KINEMATIC PROBLEMS 



323 



Kinematic Sheet No. 14 




K^onsfant Prassvr€ 

ainst 

S/Z^/ny B/ock. 




Fig. 187. 



184. Assignment. — Having given any path ARST around 
which a point is to travel, design a mechanism to guide the point, 
the mechanism to have but one rotating shaft and one rotating 
disc cam, although other machine elements may enter into the 
construction. No part of the mechanism, excepting a single 
driving arm, shall project within the path ARST, or above the 
horizontal line drawn through T. 

The movement of the point will be 

A to R ( ) of period of rotation. 

R to S ( ) of period of rotation. 

S to T ( ) of period of rotation. 

T to A ( ) of period of rotation. 



324 MACHINE DESIGN 

Original Kinematic Problems 

185. Assignment. — Sketches AtoE show some of the common 
forms of paper clips on the market. The problem is to design 
cams, connecting levers and properly shaped dies to produce 
from a spool of wire some one of the forms indicated. Sketches 
may be taken as full size. 






Fig. 188. 

186. Assignment. — The path of a block consists of two parts, 
AB and BC. BC is J the length oi AB and perpendicular to AB. 

Motion cycle to be as follows: 

\ B to A, uniform acceleration. 

f B to A, uniform motion. 

\ B io A, uniform acceleration. 

A to B, harmonic motion. 
\ B to C, uniform acceleration, 
f 5 to C, uniform motion. 
\ B to C, uniform acceleration. 

C to 5, harmonic motion. 

The motion of the block is to be obtained from a single disc 
cam, and no part of the mechanism — excepting a single guiding 
arm to impart motion to block — shall extend outside the angle 
ABD, where D is on a continuation of CB. Use not more than 
two levers or bell cranks and no connecting links, and have 
block make complete cycle in one revolution of cam. 

187. Assignment. — The path of a block is to be a square 
A, B, C, D, the block to be driven by a single cylindrical cam 
rotating with a vertical shaft, i.e., shaft is perpendicular to plane 



KINEMATIC PROBLEMS 325 

of path. No part of the driving mechanism is to operate in the 
plane of the square. The motion cycle is to be: 

i A to B, uniform acceleration. 

^ A to B, uniform motion. 

I A to B, uniform acceleration. 

This to be repeated for B to C, C to D, and D to A. 

188. Assignment. — A follower block has motion along a path 
ABCD. AB and DC are each perpendicular to BC, on the same 
side, and at the ends of the line BC. In length, these path 
parts bear the following relations: BC = 2AB = 11DC. One 
cylindrical cam is to be used, and no part of the driving mechan- 
ism is to extend within the figure ABCD, at any time during the 
motion, the cycle of which is to be : 

i B to Cj constant acceleration. 

i B to C, uniform motion. 

i B to C, constant deceleration. 

i C to D, constant acceleration. 

f C to D, uniform motion. 

^ C to D, constant deceleration. 
D to C, same variations as B to C 
C to B, same variations as B to C. 
B to A, A to B, harmonic motion. 

189. Assignment. — A follower block is to move in a groove 
whose center line is ABC. BC is perpendicular to AB, and f as 
long as AB. The motion is to be given by a single cylindrical 
cam, which may, however, carry more than one groove. Not 
more than two levers or bell cranks and not more than two 
connecting rods may be used. No part of the mechanism is to 
extend within the angle ABC, and the cam must lie in the angle 
made by prolonging AB and CB. 

I A to B, constant acceleration. 
^ A to B, constant motion. 
^ A to B, constant deceleration. 
I 5 to C, increasing harmonic. 
I B to C, constant motion. 



326 MACHINE DESIGN 

f 5 to C, decreasing harmonic. 
I C to B, constant acceleration. 
4 (7 to J5, constant motion. 
i C to B, constant deceleration. 
i B to A, increasing harmonic. 
i B to A, constant motion. 
I B to A, decreasing harmonic. 

190. Assignment. — Block as follower to move in groove whose 
center line is ABC. 

Block to be driven by two disc cams on the same shaft. 

No part of mechanism, excepting a single driving arm, to 
extend inside the angle ABC or above the line ABD. Mechanism 
to be sufficiently substantial and positive for die work. 



A- 



B 
Fig. 189. 



D 



Motion to be as follows: 



\B to A, uniform acceleration. 

\ B to A^ uniform velocity. 

\B to A^ uniform acceleration (decreasing). 

A to 5, harmonic motion (increasing and decreasing) . 

B to C, same as the first three above from B to A. 

C to 5, harmonic motion (increasing and decreasing) . 

191. Assignment. — Block as follower working in slot whose 
center line is ABC. To be driven by two cylindrical cams with 
axes at right angles to each other. No part of the mechanism 
to extend within the angle ABC. 

Motion of block to be as follows : 
\ BA to left, constant acceleration. 
\ BA to left, constant velocity. 
\ BA to left, constant acceleration. 
J AB to right, increasing and decreasing harmonic motion. 



KINEMATIC PROBLEMS 



327 



J BA to left, constant acceleration. 
J BA to left, constant velocity. 
J BA to left, constant acceleration. 

AB to right, increasing and decreasing harmonic motion. 
J BC up, constant acceleration, 
i BC up, constant velocity, 
i BC up, constant acceleration. 



^ 



Fig. 190. 
AB 



CB = 



^ BC down, harmonic motion. 
I BC up, constant acceleration. 

BC up, constant velocity. 

BC up, constant acceleration. 

CB down, increasing harmonic motion. 

CB down, constant velocity, 
f CB down, decreasing harmonic motion. 



192. Assignment. — ^Path ABC of block as follower to be a 
groove. Block to move from A to B to C to 5 to A. 



A- 



Fig. 191. 



CB 



iAB. 



B 



No cams are to be used in the mechanism, and no part of the 
driving mechanism is to extend within the angle ABC. 

Rack and pinion or chain drive cannot be used directly to 
produce motion. 



328 MACHINE DESIGN 

193. Assignment. — Block as follower to be driven in groove 
with center line ABC. No part of mechanism to extend within 
the angle ABC. Use one disc cam and one cylindrical cam. 
No bell cranks or pivoted levers can be employed. Motion of 
block to be as follows : 

J BA, constant acceleration. 

4 BA, constant velocity. 

J BA, constant acceleration. 

i AB, harmonic motion increasing. 

AB, constant velocity. 

AB, harmonic motion decreasing. 

BC, constant acceleration. 

BC, constant velocity. 

BC, constant acceleration. 

CB, harmonic motion. 



A 



B 



Fig. 192. 
CB = \AB. 

194. Assignment. — Block as follower to move in groove whose 
center line is ABC. Driven by a single cylindrical cam, which 
may, however, carry more than one groove. Not more than 
two levers or bell cranks and two connecting rods may be used. 

No part of mechanism to extend within the angle ABC and 
the cam itself must be located in the angle DBE. 

Motion of block to be as follows : 

i A to B, constant acceleration. 
J A to 5, constant velocity. 
i A to B, constant acceleration. 
I 5 to 0, increasing harmonic. 
I B to C, constant velocity, 
f 5 to C, decreasing harmonic. 
i C to B, constant acceleration. 



KINEMATIC PROBLEMS 



329 



i C to B, constant velocity. 
i C to B, constant acceleration. 
i B to A, increasing harmonic. 
i B to A, constant velocity. 
i B to A, decreasing harmonic. 



A- 



C 



B 



Fig. 193. 
BC = IAB. 



-E 



195. Assignment. — Block as follower to be driven in groove 
with center line ABC. No part of mechanism to extend within 
the angle ABC. Use one disc cam and one cylindrical cam. 
No bell cranks or pivoted levers may be employed. 



C 



B 



Fig. 194. 
CB = \AB. 

Motion to be as follows: 
\ BA, constant acceleration. 
J BA, constant velocity. 
J BA, constant acceleration (decreasing), 
f AB, harmonic motion (increasing). 
I AB, constant velocity. 
i AB, harmonic motion (decreasing). 

BC, same motion as given in the first three above for B to A. 

CB, with harmonic motion (increasing and decreasing). 



330 



MACHINE DESIGN 



196. Assignment. — Design a mechanism to drive a block over 
the path ABCDCBA. 



A 





J 


Fig. 
CT) = 
CB = 


B 
195. 
^lAB, 
-iAB, 



D 



ABCED is center line of groove. 
Use no cams, no chains, no racks. 
Single rotating shaft. 

197. Assignment. — ^Path ABC to be a groove. CB=i AB. 

Block to be driven in this groove. 

Motion to be obtained from a single disc cam and no part of 
the mechanism, excepting a single guiding arm to impart motion 
to block, to extend without the angle ABD at any time during 
the period of motion. Use not more than two levers or bell 
cranks and no connecting links. 

Blocks to make complete cycle in one revolution of cam. 



A- 



B 



D 



Fig. 196. 



Motion of block to be as follows : 



\B \jO A^ uniform acceleration, 
f 5 to A, uniform motion. 
I" B to A, uniform acceleration. 
A to B, harmonic motion. 



KINEMATIC PROBLEMS 331 

i B to C, uniform acceleration. 
J B to C, uniform motion. 
I B to C, uniform acceleration. 
C to B, harmonic motion. 

198. Assignment. — Construct cams and mechanism to drive a 
point over the path ABCDCE and reverse. Use but one rotat- 
ing shaft and not more than two cams, either disc or cylindrical. 
No part of the mechanism to extend within the angle ABD, or 
above the line DF. 



D 



A 



-E 



B 

Fig. 197. 
CB = \ AB. 
CD = i CB. 
CE = BD. 



199. Assignment. — ABC DA is the center line of a groove. 
Design a mechanism to drive a square block over the groove. 
Use one rotating shaft, 0. 

There should be no opportunity for block to wedge at corners. 
Use no cams, chains, racks or screws. 




Fig. 198. 



Following along the line of the above assignment, others may 
be made referring to a base runner around the base-ball diamond 



332 



MACHINE DESIGN 



using various forward and backward movements, and various 
constant velocities and accelerations. 

200. Assignment. — Required to design the mechanism and 
cams to produce some word at the end of the pencil arm. The 
location of the parts should be selected so as to show the univer- 




FiG. 199. 

sality of the cam motion. All dimensions are to be selected by 
the student. Care should be exercised that the cam curves do 
not slope at too great an angle. 

201. Assignment. — Required to design the mechanism and 
cams to produce some word at the end of the pencil arm. The 



KINEMATIC POBLEMS 



333 



mechanism is to have but three moving parts. All dimensions 
are to be selected by the student. Care should be exercised that 
the cam curves do not slope at too great an angle. 




Fig. 200. 



334 



MACHINE DESIGN 



202. Assignment. — Required to design the mechanism for a 
writing cam as shown in Fig. 201. The sliding block X is moved 
by screw connection. All sizes to be selected by the student. 
Such a cam may be used for outlining any simple figure in design 
as well. 




KINEMATIC PROBLEMS 
Mechanism of the Rites Inertia Governor 



335 




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336 



MACHINE DESIGN 
Mechanism of the Centrifugal Governor 




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KINEMATIC PROBLEMS 



337 



Mechanism of the Straight Line Governor 




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Cfon di Ti on « ; 

PKsnxi Utjliv Inside Aam»3aioi\ 
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207. Assignment.— In this analysis assign the lead and the cut- 
this cut-off and draw in position of cut-off. Finally draw valve 



v'i 



r 



schaert Valve Gear 

Valve Motions) 




LapCt 



VvALSCHAIiRT VMA/H GUAR 
ANALYSIS 



). 



(the latter varies from 20 to 80 per cent). Set the link to give 
pse and fill in table of events. 



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INDEX 



Abbreviations, 1 

Air hoist, 299 

Alloy steels, 14 

Alloys, 15 

Arms of gears, 196 

Automobile clutches, 176 

Ball bearings, design, 157 

endurance, 157 

journal, 153 

materials, 156 

step, 155 
Beams, cast iron, 28 

formulas, 7 

uniform strength, 8 
Bearings, adjustment, 129 

baU, 153 

experiments, 138 

friction, 134, 145 

heating, 136 

Hyatt, 160 

journal, 128 

lubrication, 131 

Mossberg, 164 

multiple, 149 

pressure, 134 

roller, 159 

sliding, 120 

step, 144 
Belting, friction, 221 

slip, 223 

speed, 227 

strength, 225 

width, 226 
Bevel shear, plain, 287 

rotary, 296 
Bobbin winder, 313 
Boiler, shells, 50 

tubes, 55 
Bolts and nuts, 01 



Brass, 16 
Bronze, 16 
Bulldozer, 293 
Butt joints, 99 

Cams, accelerating, 324 

conical, 321 

crosshead, 320 

lever, 319 

planer, 311 

sewing machine, 312 

steamboat, 314 

writing, 332 
Caps and bolts, 142 
Castings, iron, 10 

steel, 14 
Chain drives, 197 
Clip former, 324 
Clutches, 173 

automobile, 176 

press, 280 
Columns, 4 
Cotters, 104 
Cotton ropes, 231 
Coupling bolts, 178 
Couplings, 171 
Crane hooks, 94 
Cranks and levers, 200 
Crucible steel, 13 
Curved frames, design, 47 

tests, 42 
Cylinders, steam, 78 

tests, 80 

thick, 51 

Die heads, punch, 283 
sliding, 251 
stationary, 253 

Discs, conical, 216 
logarithmic, 217 



339 



340 



INDEX 



Discs, plain, 215 

tests, 218 
Drawings, size and scale, 236 

Elliptic springs, 114 

Factors of safety, 17 
Fittings, pipe, flanged, 71 

screwed, 69 
Flanged fittings, 71 
Flanging machine, 297 
Flat plates, 83 

springs, 113 
Fly wheel, experiments, 207 

press, 274 

rims, 204 

safe speeds, 205 
Formulas, 3 
Frames, curved, 42 

design, 21 

machine, 26 

press, 255 

riveter, 40 

shape, 38 

shear, 45, 266 

stresses, 39 
Friction, belting, 221 

journals, 139 

pivots, 145 

Gears, bevel, 195 

cut, 189 

design, 310 

rim and arms, 196 

speed, 212 

teeth, 186 
Governor, centrifugal, 336 

shaft, 337 

Hangers, 182 
Heating of journals, 136 
Helical springs, 107 
Hoist, air, 299 
Hooks, crane, 94 
Hoops, steel, 68 
Hyatt bearings, 160 



Iron, cast, 10 

malleable, 11 
wrought, 12 

Joint pins, 104 
Joints, rim, 210 
Joints, riveted, butt, 99 

diamond, 103 

lap, 98 

tube, 67 
Journals, 128 

strength of, 142 

Keys, shafting, 178 
Kinematics of machines, 309 

Lap joints, 98 

Lever design, 240 

Link motion, Stephenson, 338 

Walschaert, 339 
Lubrication, 131 

Machine frames, 26 

screws, 93 

supports, 25 
Malleable iron, 11 
Manganese, bronze, 16 

steel, 14 
Manila ropes, 229 
Materials, 9 
Metals, strength of, 18 
Mossberg bearings, 165 
Mushet steel, 15 

Nickel steel, 14 
Notation, 2 

Open hearth steel, 13 

Phosphor bronze, 16 
Pipe, fittings, 69 

tables, 56 
Pivots, conical, 146 

flat, 145 

Schiele's, 147 
Plates, flat, 83 

steel, 87 



INDEX 



341 



Power press, 294 
Press, foot power, 262 

hand power, 261 

power, 294 

shear, 295 

toggle joint, 235 
Pulleys for press, 273 
Punch, hand power, 263 

horizontal, 288 

Quick return, 315 

Riveted joints, 96 
Riveter, Allen, 300 

aUigator, 304 

frames, 40 

Hanna, 302 

hydrauHc, 307 

lever, 306 

mudring, 305 
Riveting machine, hydraulic, 308 
Roller bearings, 159 

design, 163 

step, 162 
Rope transmission, cotton, 231 

Manila, 229 

strength, 230 

wire, 232 
Rotary shear, 295 

Schiele's pivot, 147 
Screw, design, 246 

machine, 93 
Sections, cast iron, 27 
Shaft for press, 276 
Shafting, bending, 169 

clutches, 173 

coupUngs, 171 

diameter, 168 

hangers, 182 

keys, 178 
Shapes of frames, 38 
Shear press, clutches, 280 

die head, 283 

fly wheel, 274 

forces, 272 

frame, 45, 266 



Shear press, gears, 277 

pulleys, 273 

shaft, 276 

sliding head, 279 

types, 284 
Shear, rotary, 295 
Shells, thin, 50 

thick, 51 
Silent chains, 199 
Slides, angular, 120 

circular, 124 

flat, 122 

gibbed, 122 
Slip of belts, 223 
Springs, eUiptic, 114 

experiments, 109 

flat, 113 

hehcal, 107 

torsion, 112 
Sprocket wheels, 197 
Standard for press, 247 
Steam cyUnders, 78 
Steel, alloys, 14 

Bessemer, 13 

castings, 14 

crucible, 13 

mushet, 15 

open hearth, 13 

plates, 87 
Step bearings, 144 
Stephenson link motion, 338 
Strength of materials, 17 
Stuffing boxes, 124 
Supports, machine, 25 

Tests of gears, experiments, 191 

formulas, 190 

practice, 193 

proportions, 187 

strength, 188 
Thrust bearings, 150 
Toggle joint press, alternate design, 
260 

analysis, 239 

design No. 1, 235 

die head, 250 

frame, 255 



342 INDEX 

Toggle lever, 240 Units, 1 

screw, 246 ^^ ,. . i -.e 

/ , „ ,^ Vanadium steel, 15 
standard, 247 ' 

toggIe,f248 Walschaert link motion, 339 

Torsion springs, 112 Wire ropes, 232 

Tubes, boiler, 55 Wooden pulleys, 212 

joints, 67 Writing cam, 332 

tests on, 62 Wrought iron, 12 



MATHEMATICAL SERIES 

While this series has been planned to meet the needs of the 
student who is preparing for engineering work, it is hoped that it 
will serve equally well the purposes of those schools where mathe- 
matics is taken as an element in a liberal education. In order that 
the applications introduced may be of such character as to interest 
the general student and to train the prospective engineer in the 
kind of work which he is most likely to meet, it has been the poKcy 
of the editors to select, as joint authors of each text, a mathemati- 
cian and a trained engineer or physicist. 

The following texts are ready: 

I. Calculus. 

By E. J. TowNSEND, Profosser of Mathematics, and G. A. Good- 
enough, Professor of Thermodynamics, University of Illinois. 
$2.50. 

II. Essentials of Calculus. 
By E. J. TowNSEND and G. A. Goodenough. $2.00. 

III. College Algebra. 

By H. L. RiETZ, Assistant Professor of Mathematics, and Dr. A. 
R. Crathorne, Associate in Mathematics in the University of 
Illinois. $1.40. 

IV. Plane Trigonometry, with Trigonometric and Logarithmic 

Tables. 
By A. G. Hall, Professor of Mathematics in the University of 
Michigan, and F. H. Frink, Professor of Railway Engineering in 
the University of Oregon. $1.25. 

V. Plane and Spherical Trigonometry. 

(Without Tables) 
By A. G. Hall and F. H. Frink. $1.00. 

VI. Trigonometric and Logarithmic Tables. 
By A. G. Hall and F. H. Frink. 75 cents. 

The following are in preparation: 

Plane Analytical Geometry. 

ByL. W. DowLiNG, Assistant Professor of Mathematics, and F. E. 
TuRNEAURE, Dean of the College of Engineering in the tjniversity 
of Wisconsin. 

Solid Analytical Geometry. 

By Virgil Snyder, Professor in Cornell University, and C. H. 
SiSAM, Assistant Professor in the University of Illinois. 

Young and Schwartz's Elementary Geometry. 

By J. W. Young, Professor of Mathematics in Dartmouth Col- 
lege, and A. J. Schwartz, William McKinley High School, St. Louis. 

HENRY HOLT AND COMPANY 

NEW YORK CHICAGO 



ENGINEERING BOOKS 

Hoskins's Hydraulics. 

By L. M. HosKiNS, Professor in Leland Stanford University. 
8vo. 271 pp. $2.50. 

A comprehensive text-book, intended for the fundamental course 
in the subject usually offered in schools of engineering, but some- 
what more compact in treatment than the ordinary treatise now 
available. 

Russell's Text-book on Hydraulics. 

By George E. Russell, Assistant Professor of Civil Engineering, 
Massachusetts Institute of Technology, viii+183 pp. 8vo. $2.50. 

This book is designed primarily for classroom use rather than 
for reference for practising engineers. It avoids discussion of speci- 
alized topics which are taught separately with special books and 
devotes itself to the consideration of the more common and im- 
portant subjects. At the end of each chapter are given problems 
to illustrate the appHcation of the principles just preceding. 

Benjamin s Machine Design. 

By Charles H. Benjamin, Professor in Purdue University. 
i2mo. 202 pp. $2.00. 

Benjamin and Hoffman's Machine Design. 

By Charles H. Benjamin, Dean of the Schools of Engineering 
of Purdue University, and James D. Hoffman, Professor of Me- 
chanical Engineering and Practical Mechanics in the University of 
Nebraska. 

Slocum's Theory and Practice of Mechanics. 

By S. E. Slocum, Professor of Applied Mathematics in the Uni- 
versity of Cincinnati. , 

Goodenough s Principles of Thermodynamics. 

By G. A. Goodenough, Professor of Thermodynamics in the Uni- 
versity of Ilhnois. xiv+327 pp. Svo. $3.50. 

Leffler s The Elastic Arch. 

With special reference to the Reinforced Concrete Arch. By 
Burton R. Lepeler, Engineer of Bridges on the Lake Shore and 
Michigan Southern Railway, viii-fsypp. i2mo. $1.00. 

HENRY HOLT AND COMPANY 

NEW YORK CHICAGO 



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